Missing Terms in Sequences - ISEE Upper Level Quantitative Reasoning

A geometric sequence has the property that the ratio of any term to the previous term is constant. If the 2nd term is 6 and the 5th term is 162, what is the 7th term?
Answer: 1458 Geometric sequences are foundational in quantitative reasoning, where each term equals the previous term multiplied by a constant ratio (r). When you see a geometric sequence problem, your goal is to find this common ratio first, then use it to calculate unknown terms. To find the common ratio, use the relationship between the given terms. The 2nd term is 6 and the 5th term is 162. Since each term multiplies by r, the 5th term equals the 2nd term times $r^3$ (three steps from 2nd to 5th). So: $162 = 6 \times r^3$ , which gives us $r^3 = 27$ , therefore $r = 3$ . Now you can find the 7th term. From the 5th term (162) to the 7th term requires multiplying by $r^2$ : $162 \times 3^2 = 162 \times 9 = 1458$ . Let's examine why each wrong answer appears: (B) 1944 might result from incorrectly calculating $162 \times 12$ if you mistakenly thought the ratio was 6 instead of 3. (C) 2187 equals $3^7$ , which could tempt you if you confused the term position with the actual term value. (D) 2916 equals $162 \times 18$ , suggesting an error where you might have used $3^2 \times 2 = 18$ instead of just $3^2 = 9$ . The correct answer is (A) 1458. Strategy tip: Always find the common ratio first by setting up the relationship between known terms, then systematically multiply to reach your target term. Double-check by ensuring your ratio is consistent throughout the sequence.
In the sequence 2, 5, 10, 17, 26, 37, ..., each term can be expressed as $n^2 + 1$ where $n$ starts from a certain value. What is the 10th term of this sequence?
Answer: 101 When you encounter a sequence problem, your first step is to identify the underlying pattern or formula. Here, you're told that each term follows the form $n^2 + 1$ , so you need to determine what value of $n$ corresponds to each position in the sequence. Let's check the given terms against the formula $n^2 + 1$ : - First term: 2 = $1^2 + 1$ , so $n = 1$ - Second term: 5 = $2^2 + 1$ , so $n = 2$ - Third term: 10 = $3^2 + 1$ , so $n = 3$ - Fourth term: 17 = $4^2 + 1$ , so $n = 4$ The pattern is clear: for the $k$ th term in the sequence, $n = k$ . Therefore, the 10th term uses $n = 10$ : $10^2 + 1 = 100 + 1 = 101$ . Choice A (101) is correct. Choice B (122) would result from $n = 11$ , which corresponds to the 11th term, not the 10th. This is a common off-by-one error. Choice C (145) equals $12^2 + 1$ , representing the 12th term. Students might reach this by miscounting or making calculation errors. Choice D (170) equals $13^2 + 1$ , the 13th term. This suggests even further miscounting in the sequence. Strategy tip: In sequence problems, always verify the given formula with the provided terms first to understand the relationship between position and the variable. Then apply that same relationship to find your target term. Double-check your arithmetic, as these problems often include answer choices that result from common calculation mistakes.
A recursive sequence is defined by $a_1 = 5$ and $a_{n+1} = a_n + 2n + 2$ for $n ≥ 1$ . What is $a_4$ ?
Answer: 23 Apply the rule one step at a time, using the value of $n$ for the term you start from. $a_2 = a_1 + 2(1) + 2 = 5 + 4 = 9$ $a_3 = a_2 + 2(2) + 2 = 9 + 6 = 15$ $a_4 = a_3 + 2(3) + 2 = 15 + 8 = 23$ So $a_4 = 23$ . The value 21 comes from adding only $2n$ in the last step (15 + 6) and forgetting the extra 2. The value 19 comes from reusing $n = 2$ in the last step (15 + 4). The value 17 comes from using $n = 1$ in every step, adding 4 three times (5 + 4 + 4 + 4).
The sequence $\frac{1}{3}, \frac{1}{6}, \frac{1}{12}, \frac{1}{24}, ...$ is geometric. What is the 7th term?
Answer: $\frac{1}{192}$ When you encounter a geometric sequence problem, you need to identify the first term and common ratio, then use the formula for the nth term: $a_n = a_1 \cdot r^{n-1}$ . Looking at this sequence $\frac{1}{3}, \frac{1}{6}, \frac{1}{12}, \frac{1}{24}, ...$ , the first term is $a_1 = \frac{1}{3}$ . To find the common ratio, divide any term by the previous term: $r = \frac{1/6}{1/3} = \frac{1}{6} \cdot \frac{3}{1} = \frac{1}{2}$ . You can verify this works for other consecutive terms. For the 7th term, substitute into the formula: $a_7 = \frac{1}{3} \cdot \left(\frac{1}{2}\right)^{7-1} = \frac{1}{3} \cdot \left(\frac{1}{2}\right)^6 = \frac{1}{3} \cdot \frac{1}{64} = \frac{1}{192}$ . Looking at the wrong answers: Choice B ( $\frac{1}{256}$ ) results from calculating $\left(\frac{1}{2}\right)^8$ , suggesting confusion about which power to use. Choice C ( $\frac{1}{384}$ ) comes from multiplying the correct answer by 2, possibly from using $\left(\frac{1}{2}\right)^7$ instead of $\left(\frac{1}{2}\right)^6$ . Choice D ( $\frac{1}{768}$ ) appears to result from incorrect arithmetic when combining fractions. The correct answer is A. Remember that in the geometric sequence formula $a_n = a_1 \cdot r^{n-1}$ , the exponent is always one less than the term number you're seeking. This is the most common error students make with geometric sequences, so double-check that you're using $n-1$ as your exponent.
A sequence is given by $a_n = 3n^2 - 2n + 1$ . What is the value of $a_6 - a_4$ ?
Answer: 56 When you encounter sequence problems with explicit formulas like $a_n = 3n^2 - 2n + 1$ , you need to substitute the given values and then perform the requested operation. To find $a_6 - a_4$ , you must calculate each term separately first. For $a_6$ , substitute $n = 6$ : $a_6 = 3(6)^2 - 2(6) + 1 = 3(36) - 12 + 1 = 108 - 12 + 1 = 97$ For $a_4$ , substitute $n = 4$ : $a_4 = 3(4)^2 - 2(4) + 1 = 3(16) - 8 + 1 = 48 - 8 + 1 = 41$ Therefore: $a_6 - a_4 = 97 - 41 = 56$ Looking at the wrong answers: Choice B (58) likely results from a calculation error, perhaps in computing one of the squares or in the final subtraction. Choice C (60) might come from incorrectly calculating $a_6$ as 101 instead of 97, possibly by making an arithmetic mistake with $3 \times 36$ . Choice D (62) could result from multiple computational errors or from mishandling the negative terms in the formula. The key strategy here is methodical substitution and careful arithmetic. Don't try to find shortcuts by factoring or looking for patterns until you're completely comfortable with direct substitution. Double-check your arithmetic at each step, especially when dealing with squares and negative terms, as these are common sources of errors on standardized tests.
A fundraiser's totals rise steadily; find the missing term: 120, 150, $\underline{\hspace{2em}}$ , 210, 240?
Answer: 180 This question tests upper-level ISEE quantitative reasoning skills, specifically determining missing terms in sequences. Sequences follow specific patterns, such as arithmetic or geometric, where each term depends on the previous term(s) based on a set rule. In this fundraiser sequence, 120, 150, _, 210, 240, the pattern involves adding 30 to each term to get the next term (120+30=150, 150+30=180, 180+30=210, 210+30=240). Choice C (180) is correct because it follows the identified arithmetic pattern of adding 30, showing understanding of constant growth in real-world contexts. Choice B (170) is incorrect because it assumes a different increment, perhaps thinking the increase is 20 rather than 30, a common error when not carefully calculating differences. To help students: Connect sequences to real-world scenarios like fundraising to make patterns more meaningful, always calculate differences between known consecutive terms, and verify patterns work for all given values before selecting an answer.
What is the missing term in the sequence: 1, 4, 9, $\underline{\hspace{2em}}$ , 25?
Answer: 16 This question tests upper-level ISEE quantitative reasoning skills, specifically determining missing terms in sequences. Sequences follow specific patterns, such as arithmetic or geometric, where each term depends on the previous term(s) based on a set rule. In this sequence, 1, 4, 9, _, 25, the pattern involves perfect squares: 1²=1, 2²=4, 3²=9, 4²=16, 5²=25. Choice C (16) is correct because it equals 4², following the identified pattern of consecutive perfect squares, showing understanding of quadratic sequences. Choice B (14) is incorrect because it assumes a different pattern, perhaps thinking the differences increase linearly rather than recognizing the square number pattern. To help students: Memorize common sequences like perfect squares (1, 4, 9, 16, 25, 36...), look for patterns in both the terms and their positions, and recognize that some sequences are based on mathematical operations applied to the term's position number.
Calculate the missing number in the sequence: 81, 27, $\underline{\hspace{2em}}$ , 3, 1?
Answer: 9 This question tests upper-level ISEE quantitative reasoning skills, specifically determining missing terms in sequences. Sequences follow specific patterns, such as arithmetic or geometric, where each term depends on the previous term(s) based on a set rule. In this sequence, 81, 27, _, 3, 1, the pattern involves dividing each term by 3 to get the next term (81÷3=27, 27÷3=9, 9÷3=3, 3÷3=1). Choice D (9) is correct because it follows the identified geometric pattern of dividing by 3, showing understanding of decreasing geometric sequences. Choice C (8) is incorrect because it doesn't maintain the consistent ratio of 1/3 between consecutive terms, perhaps resulting from a calculation error or misidentifying the pattern. To help students: Recognize that geometric sequences can involve division (multiplication by fractions), practice with both increasing and decreasing geometric sequences, and always verify the pattern by checking ratios between all consecutive terms.
A sequence follows the pattern where the $n$ th term is given by $a_n = \frac{(-1)^n}{n+1}$ . What is the sum of the 5th and 6th terms?
Answer: $-\frac{1}{42}$ When you encounter a sequence with an alternating sign pattern like $(-1)^n$ , you're dealing with terms that switch between positive and negative based on whether $n$ is even or odd. To find the sum of the 5th and 6th terms, calculate each term using the formula $a_n = \frac{(-1)^n}{n+1}$ . For the 5th term: $a_5 = \frac{(-1)^5}{5+1} = \frac{-1}{6} = -\frac{1}{6}$ For the 6th term: $a_6 = \frac{(-1)^6}{6+1} = \frac{1}{7}$ Now add them: $a_5 + a_6 = -\frac{1}{6} + \frac{1}{7}$ To add these fractions, find a common denominator: $-\frac{1}{6} + \frac{1}{7} = -\frac{7}{42} + \frac{6}{42} = -\frac{1}{42}$ This confirms answer A is correct. Answer B ( $\frac{1}{42}$ ) results from a sign error—likely forgetting that $(-1)^5 = -1$ , making the 5th term positive instead of negative. Answer C ( $-\frac{1}{21}$ ) comes from incorrectly using 21 as the common denominator instead of 42, or miscalculating $6 \times 7$ . Answer D ( $\frac{1}{21}$ ) combines both the sign error and the denominator mistake. Remember that $(-1)^n$ equals $-1$ when $n$ is odd and $+1$ when $n$ is even. Always double-check your arithmetic when adding fractions—find the least common multiple of the denominators carefully.
In the sequence $2, 6, 18, 54, 162, ...$ each term is obtained by multiplying the previous term by 3. If this pattern continues, which term will first exceed 10,000?
Answer: 9th term This question tests your understanding of geometric sequences, where each term is found by multiplying the previous term by a constant ratio. When you see a pattern like this, you need to find the general formula and then determine which term reaches the target value. In this sequence, each term equals the previous term times 3, so the common ratio is 3. The general formula for the nth term is: $a_n = 2 \times 3^{n-1}$ , where 2 is the first term. To find when this first exceeds 10,000, you need to solve: $2 \times 3^{n-1} > 10,000$ , which gives us $3^{n-1} > 5,000$ . Let's calculate the terms systematically: 8th term = $2 \times 3^7 = 2 \times 2,187 = 4,374$ , and 9th term = $2 \times 3^8 = 2 \times 6,561 = 13,122$ . Since 13,122 > 10,000, the 9th term is the first to exceed 10,000. Choice A (8th term) falls short because 4,374 < 10,000. Choice C (10th term) would be correct if the question asked for the second term to exceed 10,000, since that equals 39,366. Choice D (11th term) represents an even later term that's unnecessary since we find our answer earlier. When working with geometric sequences, always write out the general formula first, then calculate terms systematically rather than trying to guess. This prevents calculation errors and helps you see the pattern clearly.
In an arithmetic sequence, the first term is 7 and the common difference is -3. Which term in the sequence is equal to -20?
Answer: 10th term When you encounter arithmetic sequence problems, you're working with a pattern where each term differs from the previous by a constant amount. Here, you need to find which term equals -20 when the first term is 7 and the common difference is -3. The formula for the nth term of an arithmetic sequence is: $a_n = a_1 + (n-1)d$ , where $a_1$ is the first term, $d$ is the common difference, and $n$ is the term number. Setting up the equation with your given values: $-20 = 7 + (n-1)(-3)$ Solving step by step: $-20 = 7 - 3(n-1)$ $-20 = 7 - 3n + 3$ $-20 = 10 - 3n$ $-30 = -3n$ $n = 10$ So -20 is the 10th term, making (B) correct. Let's check why the other answers are wrong. For (A) the 9th term: $a_9 = 7 + 8(-3) = 7 - 24 = -17$ . For (C) the 11th term: $a_{11} = 7 + 10(-3) = 7 - 30 = -23$ . For (D) the 12th term: $a_{12} = 7 + 11(-3) = 7 - 33 = -26$ . Each of these gives you a different value than -20. Study tip: Always substitute your answer back into the original formula to verify. Also, remember that negative common differences create decreasing sequences, so larger term numbers yield smaller values. This can help you estimate whether your answer makes sense before calculating.
Identify the missing number in the sequence given the rule multiply by 3: 2, 6, $\underline{\hspace{2em}}$ , 54, 162?
Answer: 18 This question tests upper-level ISEE quantitative reasoning skills, specifically determining missing terms in sequences. Sequences follow specific patterns, such as arithmetic or geometric, where each term depends on the previous term(s) based on a set rule. In this sequence with the rule multiply by 3: 2, 6, _, 54, 162, each term is multiplied by 3 to get the next term (2×3=6, 6×3=18, 18×3=54, 54×3=162). Choice B (18) is correct because it follows the given geometric pattern of multiplying by 3, showing understanding of applying stated rules to sequences. Choice C (20) is incorrect because it doesn't follow the multiply-by-3 rule, perhaps resulting from adding rather than multiplying or misunderstanding the pattern. To help students: When a rule is explicitly given, apply it consistently to each term, verify by checking that the rule produces all given terms, and practice with both arithmetic and geometric sequences to distinguish between additive and multiplicative patterns.
A sequence begins with 1, 3, 6, 10, 15, ... where each term represents triangular numbers (sums of consecutive integers starting from 1). What is the 12th triangular number?
Answer: 78 When you encounter a sequence involving "triangular numbers," you're dealing with a specific pattern where each term represents the sum of consecutive positive integers starting from 1. The sequence 1, 3, 6, 10, 15... follows this pattern: 1, 1+2, 1+2+3, 1+2+3+4, 1+2+3+4+5... To find the 12th triangular number, you need the sum $1+2+3+...+12$ . Rather than adding all these numbers individually, use the triangular number formula: $T_n = \frac{n(n+1)}{2}$ , where $n$ is the position in the sequence. For the 12th triangular number: $T_{12} = \frac{12(12+1)}{2} = \frac{12 \times 13}{2} = \frac{156}{2} = 78$ . This confirms that B) 78 is correct. Looking at the wrong answers: A) 66 would be the result if you mistakenly used $n = 11$ instead of 12, giving you $\frac{11 \times 12}{2} = 66$ . C) 91 comes from using $n = 13$ , calculating $\frac{13 \times 14}{2} = 91$ . D) 105 results from using $n = 14$ , giving $\frac{14 \times 15}{2} = 105$ . These are all "off-by-one" or "off-by-two" errors that occur when you miscount the position. Remember the triangular number formula $T_n = \frac{n(n+1)}{2}$ — it's much faster than adding consecutive integers manually. Always double-check that you're using the correct value of $n$ for the position you want.
In an arithmetic sequence, the sum of the 3rd and 7th terms is 40, and the sum of the 5th and 9th terms is 56. What is the 10th term?
Answer: 40 Arithmetic sequences are foundational in algebra, where each term increases by the same constant difference. When you see problems involving multiple terms and their sums, you'll need to set up equations using the general formula for the nth term: $a_n = a_1 + (n-1)d$ , where $a_1$ is the first term and $d$ is the common difference. Let's work with what we know. The 3rd term is $a_3 = a_1 + 2d$ and the 7th term is $a_7 = a_1 + 6d$ . Since their sum is 40: $a_3 + a_7 = (a_1 + 2d) + (a_1 + 6d) = 2a_1 + 8d = 40$ . Similarly, $a_5 + a_9 = (a_1 + 4d) + (a_1 + 8d) = 2a_1 + 12d = 56$ . Now you have two equations: $2a_1 + 8d = 40$ and $2a_1 + 12d = 56$ . Subtracting the first from the second: $4d = 16$ , so $d = 4$ . Substituting back: $2a_1 + 8(4) = 40$ , which gives $a_1 = 4$ . The 10th term is $a_{10} = 4 + 9(4) = 40$ . Choice A (34) would result from miscalculating the common difference as 3 instead of 4. Choice B (36) occurs if you confuse the 9th term for the 10th. Choice C (38) comes from an arithmetic error when finding $a_1$ , likely calculating it as 2 instead of 4. Remember: when working with arithmetic sequences, always double-check your system of equations and verify your common difference by substituting back into the original conditions.
The sequence 3, 7, 15, 31, 63, ... follows the pattern $a_n = 2^{n+1} - 1$ . What is the 8th term?
Answer: 511 When you encounter a sequence with an explicit formula, you need to substitute the term number into the given pattern to find that specific term. Given the pattern $a_n = 2^{n+1} - 1$ , you can find the 8th term by substituting $n = 8$ : $a_8 = 2^{8+1} - 1 = 2^9 - 1$ Calculate $2^9 = 512$ , so $a_8 = 512 - 1 = 511$ . You can verify this makes sense by checking the pattern with the given terms. For the first term: $a_1 = 2^{1+1} - 1 = 2^2 - 1 = 4 - 1 = 3$ ✓. For the second term: $a_2 = 2^{2+1} - 1 = 2^3 - 1 = 8 - 1 = 7$ ✓. Choice A (255) would result from calculating $2^8 - 1 = 256 - 1 = 255$ . This is the trap of forgetting to add 1 to the exponent—using $2^n - 1$ instead of $2^{n+1} - 1$ . Choice C (1023) comes from $2^{10} - 1 = 1024 - 1 = 1023$ . This error occurs when you mistakenly use $n + 2$ in the exponent instead of $n + 1$ . Choice D (2047) results from $2^{11} - 1 = 2048 - 1 = 2047$ , which would be the 10th term, not the 8th. The correct answer is B (511). Strategy tip: Always double-check explicit formulas by testing them against given sequence values first. This catches formula misreading errors before you invest time in calculations with the wrong pattern.
Consider the alternating sequence: 1, -4, 9, -16, 25, -36, ... What is the 11th term?
Answer: 121 When you encounter a sequence problem, start by identifying the pattern in both the magnitude and the signs of the terms. This sequence involves two separate patterns working together. First, look at the absolute values: 1, 4, 9, 16, 25, 36... These are perfect squares: $1^2, 2^2, 3^2, 4^2, 5^2, 6^2$ . So the $n$ th term has magnitude $n^2$ . Next, examine the signs: positive, negative, positive, negative, positive, negative... The pattern alternates, with odd-positioned terms being positive and even-positioned terms being negative. This means the $n$ th term equals $(-1)^{n+1} \cdot n^2$ . For the 11th term: $(-1)^{11+1} \cdot 11^2 = (-1)^{12} \cdot 121 = 1 \cdot 121 = 121$ . Since 12 is even, $(-1)^{12} = 1$ , making the 11th term positive. Choice A (-121) represents the mistake of thinking the 11th term should be negative because you might incorrectly use $(-1)^{11}$ instead of $(-1)^{11+1}$ . Choice C (-100) combines two errors: using $10^2$ instead of $11^2$ and applying the wrong sign. Choice D (100) uses the wrong base (10 instead of 11) but gets the sign correct. The answer is B (121). Remember: in alternating sequences, carefully track which formula governs the sign pattern. Test your sign rule with the first few terms to verify you have it right before applying it to later terms.
Find the value that completes the sequence: 3, 6, 12, $\underline{\hspace{2em}}$ , 48?
Answer: 24 This question tests upper-level ISEE quantitative reasoning skills, specifically determining missing terms in sequences. Sequences follow specific patterns, such as arithmetic or geometric, where each term depends on the previous term(s) based on a set rule. In this sequence, 3, 6, 12, _, 48, the pattern involves multiplying each term by 2 to get the next term (3×2=6, 6×2=12, 12×2=24, 24×2=48). Choice D (24) is correct because it follows the identified geometric pattern of doubling, showing understanding of multiplicative sequences. Choice C (22) is incorrect because it assumes an arithmetic pattern rather than geometric, a common error when students don't test whether differences or ratios are constant. To help students: Always check both differences and ratios between terms, practice recognizing geometric sequences where each term is multiplied by the same factor, and verify the pattern works for all given terms.
The sequence $1, 1, 2, 3, 5, 8, 13, ...$ follows the Fibonacci pattern where each term after the second is the sum of the two preceding terms. What is the 12th term?
Answer: 144 When you encounter a Fibonacci sequence problem, you need to systematically calculate each term using the defining rule: each term equals the sum of the two preceding terms. Starting with the given sequence $1, 1, 2, 3, 5, 8, 13, ...$ , let's continue calculating to find the 12th term: - 1st term: 1 - 2nd term: 1 - 3rd term: $1 + 1 = 2$ - 4th term: $1 + 2 = 3$ - 5th term: $2 + 3 = 5$ - 6th term: $3 + 5 = 8$ - 7th term: $5 + 8 = 13$ - 8th term: $8 + 13 = 21$ - 9th term: $13 + 21 = 34$ - 10th term: $21 + 34 = 55$ - 11th term: $34 + 55 = 89$ - 12th term: $55 + 89 = 144$ The 12th term is 144, making (B) correct. Looking at the wrong answers: (A) 89 is actually the 11th term—a common error if you miscount positions or stop calculating one step too early. (C) 233 would be the 13th term ( $89 + 144 = 233$ ), showing what happens when you go one step too far. (D) 377 is the 14th term, indicating even more overcalculation. Strategy tip: For Fibonacci problems, always write out each term systematically and double-check your position counting. The most common errors come from miscounting which term you're looking for or making arithmetic mistakes in the addition sequence.
Consider the sequence: $\frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \frac{4}{5}, ...$ What is the 15th term expressed as a fraction in lowest terms?
Answer: $\frac{15}{16}$ When you encounter a sequence problem, your first step is identifying the pattern by examining how each term relates to its position in the sequence. Looking at the given terms: $\frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \frac{4}{5}, ...$ , notice that each fraction follows a clear pattern. The numerator equals the term's position number (n), while the denominator equals n + 1. So the general formula for the nth term is $\frac{n}{n+1}$ . To find the 15th term, substitute n = 15 into this formula: $\frac{15}{15+1} = \frac{15}{16}$ . Since 15 and 16 share no common factors (15 = 3 × 5, and 16 = 2⁴), this fraction is already in lowest terms. Looking at the wrong answers: Choice B gives $\frac{14}{15}$ , which would be the 14th term using our formula. This represents the common error of calculating the wrong position. Choice C gives $\frac{16}{17}$ , which would be the 16th term—another position error. Choice D gives $\frac{13}{14}$ , the 13th term, showing yet another miscounting mistake. The correct answer is A: $\frac{15}{16}$ . Strategy tip: For sequence problems, always write out the general formula first by comparing several given terms. Then double-check which term position the question asks for—position errors are the most common trap on these problems. Verify your pattern works for at least the first three given terms before applying it.
What term should replace the underscore to continue the sequence: 2, 3, 5, $\underline{\hspace{2em}}$ , 13?
Answer: 8 This question tests upper-level ISEE quantitative reasoning skills, specifically determining missing terms in sequences. Sequences follow specific patterns, such as arithmetic or geometric, where each term depends on the previous term(s) based on a set rule. In this sequence, 2, 3, 5, _, 13, the pattern involves adding consecutive terms to get the next term (2+3=5, 3+5=8, 5+8=13), which is the Fibonacci pattern. Choice B (8) is correct because it follows the identified pattern of adding the two previous terms, showing understanding of recursive sequences. Choice C (9) is incorrect because it assumes a different pattern, perhaps thinking the differences increase by 1 each time, missing the Fibonacci relationship. To help students: Look for patterns involving relationships between multiple terms, practice recognizing famous sequences like Fibonacci, and always verify the pattern by checking if it produces all given terms correctly.

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