Identify and extend numeric and algebraic patterns - TACHS Math

1. The sequence $2,\;5,\;14,\;41,\;?$ follows the rule "multiply by 3, then subtract 1." What number should replace the question mark?
   Answer: 122 When you encounter sequence problems with a given rule, your job is to apply that rule step-by-step to verify the pattern and find the next term. Let's trace through this sequence using the rule "multiply by 3, then subtract 1": Starting with 2: $2 \times 3 - 1 = 6 - 1 = 5$ ✓ Starting with 5: $5 \times 3 - 1 = 15 - 1 = 14$ ✓ Starting with 14: $14 \times 3 - 1 = 42 - 1 = 41$ ✓ Starting with 41: $41 \times 3 - 1 = 123 - 1 = 122$ The next term is 122, making B the correct answer. Looking at the wrong choices: A) 121 represents a common error where you might subtract 1 first, then multiply by 3: $(41-1) \times 3 = 120$, or perhaps miscalculate $41 \times 3 = 122$ as 121. C) 123 occurs when you forget the "subtract 1" step and just multiply: $41 \times 3 = 123$. D) 124 might result from adding 1 instead of subtracting: $41 \times 3 + 1 = 124$. For sequence problems, always write out each step of the given rule carefully and in the correct order. Double-check your arithmetic, especially with multi-step operations. The most common trap is switching the order of operations or forgetting one of the steps entirely.
2. What is the next number in the sequence $2,\;5,\;10,\;17,\;26,\;?$
   Answer: 37 When you encounter a number sequence problem, you need to look for the pattern that connects consecutive terms. Don't just look at the differences between terms—sometimes you need to dig deeper. Let's examine the differences between consecutive terms: $5-2=3$, $10-5=5$, $17-10=7$, $26-17=9$. The first differences are $3, 5, 7, 9$. Notice these are consecutive odd numbers! This means the next difference should be $11$. To find the next term in the original sequence, add $11$ to $26$: $26+11=37$. Let's verify this pattern holds: we're adding consecutive odd numbers ($3, 5, 7, 9, 11, ...$ ) to get from one term to the next. Starting with $2$: $2+3=5$, $5+5=10$, $10+7=17$, $17+9=26$, $26+11=37$. Choice (A) $34$ would result from adding $8$ to $26$, but $8$ is even and doesn't fit the consecutive odd pattern. Choice (C) $38$ would come from adding $12$, which skips $11$ in our sequence. Choice (D) $41$ results from adding $15$, which jumps too far ahead in the odd number sequence. The correct answer is (B) $37$. Strategy tip: For sequence problems, always check the differences between terms first. If those differences don't show an obvious pattern, look at the differences of the differences (called second differences). Many TACHS sequence problems involve arithmetic or geometric progressions hidden in this layered structure.
3. A sequence is defined recursively by $c_1=2$ and $c_{n+1}=3c_n+1$ for $n\ge1$. What is the value of $c_4$?
   Answer: 67 When you encounter a recursive sequence, you need to use the given formula repeatedly, building each term from the previous one. Here you start with $c_1 = 2$ and use $c_{n+1} = 3c_n + 1$ to find subsequent terms. Let's calculate step by step: - $c_1 = 2$ (given) - $c_2 = 3c_1 + 1 = 3(2) + 1 = 7$ - $c_3 = 3c_2 + 1 = 3(7) + 1 = 22$ - $c_4 = 3c_3 + 1 = 3(22) + 1 = 67$ Therefore, $c_4 = 67$, which is choice C. Looking at the wrong answers: Choice A (22) is actually $c_3$, suggesting you stopped one step too early. Choice B (40) might result from incorrectly calculating $c_3$ as 13 instead of 22, then finding $3(13) + 1 = 40$. Choice D (82) could come from a calculation error in the final step, perhaps computing $3(22) + 16$ instead of $3(22) + 1$. The key strategy for recursive sequences is to work systematically through each term without skipping steps. Write out each calculation clearly to avoid arithmetic errors, and double-check that you're applying the recursive formula correctly at each step. Remember that $n$ represents the subscript of the term you're starting with, so $c_{n+1} = 3c_n + 1$ means "the next term equals 3 times the current term plus 1."
4. What is the next perfect square after the sequence $1,\;4,\;9,\;16,\;25$?
   Answer: 36 Perfect squares are numbers that result from multiplying an integer by itself. When you see a sequence like this, you need to identify the pattern and continue it logically. Looking at the given sequence: $1, 4, 9, 16, 25$, these are the squares of consecutive integers. Specifically: $1^2 = 1$, $2^2 = 4$, $3^2 = 9$, $4^2 = 16$, and $5^2 = 25$. To find the next perfect square, you simply continue this pattern with $6^2 = 36$. You can verify this by recognizing that perfect squares follow a specific pattern in their differences. The gaps between consecutive perfect squares increase by 2 each time: from 1 to 4 is +3, from 4 to 9 is +5, from 9 to 16 is +7, from 16 to 25 is +9. The next gap should be +11, so $25 + 11 = 36$. Choice A (30) isn't a perfect square since no integer multiplied by itself equals 30. Choice B (32) is $2^5$ but not a perfect square. Choice C (35) also has no integer square root. These numbers might seem reasonable if you're just adding a constant amount to 25, but that's not how perfect squares work. Choice D (36) is correct because $6^2 = 36$. Remember that perfect squares grow at an accelerating rate, not by constant addition. When working with sequences, always identify the underlying mathematical relationship rather than looking for simple arithmetic patterns.
5. The sequence $3^0,\;3^1,\;3^2,\;3^3,\dots$ continues by repeatedly multiplying by 3. What is the 6th term?
   Answer: 243 When you encounter a sequence problem involving powers, you're working with exponential patterns where each term follows a specific rule. This sequence starts with $3^0$ and continues by multiplying each term by 3, which means you're looking at consecutive powers of 3. To find the 6th term, first identify what power of 3 it represents. Since the sequence begins with $3^0$ as the 1st term, $3^1$ as the 2nd term, and so on, the 6th term will be $3^5$. Calculate this step by step: $3^5 = 3 \times 3 \times 3 \times 3 \times 3 = 9 \times 9 \times 3 = 81 \times 3 = 243$. Choice A (81) represents $3^4$, which would be the 5th term in this sequence. This is a common off-by-one error that happens when students miscalculate which position corresponds to which power. Choice B (162) equals $2 \times 81 = 2 \times 3^4$, which isn't a power of 3 at all. This might result from computational errors or confusing the multiplication pattern. Choice D (729) represents $3^6$, which would be the 7th term. Students often get this when they assume the sequence starts with $3^1$ instead of $3^0$, shifting every term up by one power. Remember that sequence problems require careful attention to the starting position. When a sequence begins with $3^0$, the nth term will be $3^{n-1}$, not $3^n$. Always double-check whether you're counting positions correctly from the given starting point.
6. Fill in the blank to extend the pattern of fractions $\tfrac12,\;\tfrac23,\;\tfrac34,\;\tfrac45,\;\underline{\hspace{2em}}$.
   Answer: $\tfrac56$ When you encounter a fraction sequence like this, you're looking at a pattern problem that tests your ability to identify mathematical relationships. The key is to examine both the numerators and denominators separately to find their individual patterns. Looking at the given sequence $\frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \frac{4}{5}$, notice that the numerators follow the pattern 1, 2, 3, 4, while the denominators follow 2, 3, 4, 5. Each numerator increases by 1, and each denominator also increases by 1. More importantly, each denominator is exactly one more than its corresponding numerator. Following this pattern, the next fraction should have a numerator of 5 and a denominator of 6, giving us $\frac{5}{6}$. Choice A ($\frac{5}{6}$) correctly continues both patterns and maintains the relationship where the denominator equals the numerator plus one. Choice B ($\frac{5}{7}$) has the correct numerator but skips to 7 in the denominator, breaking the consecutive integer pattern. Choice C ($\frac{6}{5}$) reverses the numerator and denominator, creating an improper fraction that doesn't follow the established pattern of proper fractions less than 1. Choice D ($\frac{6}{7}$) incorrectly advances both numerator and denominator by 2 instead of 1. For pattern problems on the TACHS, always check multiple elements: individual number sequences, relationships between parts, and overall structure. Don't just focus on one aspect—look for consistency across all components of the pattern.
7. Let $b_n$ be a sequence where $b_n = 3n+2$. What is $b_7$?
   Answer: 23 When you encounter a sequence problem, you're working with a formula that generates terms based on their position. Here, the sequence $b_n = 3n + 2$ tells you exactly how to find any term: multiply the position number by 3, then add 2. To find $b_7$, substitute $n = 7$ into the formula: $b_7 = 3(7) + 2 = 21 + 2 = 23$. This gives us answer choice B. Let's examine why the other options are incorrect. Choice A (21) represents what you'd get if you forgot to add the constant term—this would be $3(7) = 21$, but you're missing the "+2" part of the formula. Choice C (25) might result from an arithmetic error, perhaps calculating $3(7) + 4 = 25$ instead of adding 2. Choice D (27) could come from multiplying incorrectly, like $3(9) = 27$, suggesting you used $n = 9$ instead of $n = 7$. The key strategy for sequence problems is careful substitution and arithmetic. Always double-check that you're using the correct value of $n$ and following the formula exactly as written. Remember that $b_n$ means "the $n$th term," so $b_7$ specifically asks for the 7th term. Write out each step: substitute the position number, multiply first, then add or subtract any constants. This systematic approach prevents the small errors that create those tempting wrong answer choices.
8. A sequence of algebraic terms is $x+2,\;3x+4,\;5x+6,\;7x+8,\;\dots$ What is the coefficient of $x$ in the 6th term of this pattern?
   Answer: 11 When you encounter a sequence problem, look for patterns in how the terms are structured. Here, each term has the form (coefficient)$x$ + (constant), so you need to identify how both parts change from term to term. Let's examine the given sequence: $x+2,\;3x+4,\;5x+6,\;7x+8,\;\dots$ Focus on the coefficients of $x$: 1, 3, 5, 7, ... These are consecutive odd numbers, increasing by 2 each time. To find the $n$th odd number, use the formula $2n-1$. Let's verify: 1st term gives $2(1)-1=1$, 2nd term gives $2(2)-1=3$, 3rd term gives $2(3)-1=5$, and 4th term gives $2(4)-1=7$. Perfect! For the 6th term, the coefficient of $x$ is $2(6)-1=12-1=11$. Looking at the wrong answers: Choice A (9) would be the coefficient if you mistakenly thought the pattern was adding 2 starting from 1, then miscounted (1, 3, 5, 7, 9 would only get you to the 5th term). Choice B (10) might result from confusing this with an even number sequence or arithmetic errors. Choice D (12) comes from forgetting to subtract 1 in the formula $2n-1$, using just $2n=2(6)=12$. The correct answer is C (11). Strategy tip: For sequence problems, always write out the first few terms to identify the pattern, then create a formula. Double-check by substituting back into known terms before finding your answer.
9. For the sequence defined by $a_n = 4n-1,$ what is the value of $a_{15}?$
   Answer: 59 When you encounter a sequence problem with an explicit formula like $a_n = 4n-1$, you're being asked to substitute a specific value for $n$ and evaluate the expression. This tests your ability to work with algebraic expressions and follow order of operations. To find $a_{15}$, substitute $n = 15$ into the formula: $a_{15} = 4(15) - 1 = 60 - 1 = 59$ So the correct answer is C) 59. Let's examine why the other choices appear. Choice A) 55 likely results from the error $4(15-1) = 4(14) = 56$, then subtracting 1 to get 55 — this incorrectly applies the subtraction before multiplication. Choice B) 56 comes from computing $4(15-1) = 4(14) = 56$, which mistakes the formula as $a_n = 4(n-1)$ instead of $a_n = 4n-1$. Choice D) 61 results from adding instead of subtracting: $4(15) + 1 = 60 + 1 = 61$. For sequence problems, always substitute carefully and follow order of operations precisely. Multiplication comes before addition and subtraction, so $4n - 1$ means "multiply 4 by $n$, then subtract 1." Double-check by working through the arithmetic step-by-step rather than trying to do it all mentally, as small errors in substitution or calculation create tempting wrong answers.
10. A Fibonacci-like sequence starts $6,\;8,\;14,\;22,\;?$ where each term is the sum of the two preceding terms. What number belongs in the blank?
    Answer: 36 When you encounter a sequence problem where "each term is the sum of the two preceding terms," you're dealing with a Fibonacci-like pattern. The key is systematically applying this rule to find each missing term. Let's trace through the given sequence: $6, 8, 14, 22, ?$ First, verify the pattern works with the given terms: - Second term (8) + Third term (14) = Fourth term: $8 + 14 = 22$ ✓ - First term (6) + Second term (8) = Third term: $6 + 8 = 14$ ✓ Now apply the same rule to find the fifth term: - Third term (14) + Fourth term (22) = Fifth term: $14 + 22 = 36$ Looking at the wrong answers: Choice A (28) might result from incorrectly adding $6 + 22$ (the first and fourth terms), but the rule specifically requires consecutive preceding terms. Choice B (30) could come from mistakenly adding $8 + 22$ (second and fourth terms), again skipping the required consecutive pattern. Choice C (34) might result from calculation errors or misunderstanding which terms to add together. The correct answer is D (36). Strategy tip: In Fibonacci-like sequences, always double-check by working backwards—verify that your answer minus the previous term equals the term before that. Here: $36 - 22 = 14$, which matches our third term. This backward verification catches most calculation errors and confirms you're applying the pattern correctly.
11. Consider the pattern $100,\;95,\;90,\;82,\;74,\;?$ where the subtractions follow the rule "subtract 5 twice, subtract 8 twice, subtract 11 twice, …." What is the next term?
    Answer: 63 When you encounter a sequence problem with changing differences, you need to identify the underlying pattern governing how the differences themselves change. Let's trace through this sequence step by step. Starting with $100, 95, 90, 82, 74$, the differences are: - $100 - 95 = -5$ (subtract 5, first time) - $95 - 90 = -5$ (subtract 5, second time) - $90 - 82 = -8$ (subtract 8, first time) - $82 - 74 = -8$ (subtract 8, second time) The pattern shows we subtract each number twice before moving to the next: subtract 5 twice, then subtract 8 twice, then subtract 11 twice. Notice that $5, 8, 11$ increases by 3 each time. Following this rule, we need to subtract 11 twice from where we left off. We've already done $82 - 8 = 74$, so now we apply the first subtraction of 11: $74 - 11 = 63$. Looking at the wrong answers: B) 64 might result from subtracting 10 instead of 11, missing the pattern increment. C) 65 could come from subtracting only 9, or miscounting the pattern. D) 68 might result from subtracting just 6, possibly confusing this with a simpler arithmetic sequence. The correct answer is A) 63. Strategy tip: For complex sequence problems, always write out the differences between consecutive terms first. Look for patterns in how these differences change—they often follow their own arithmetic or geometric progression.
12. The sequence $1,\;3,\;6,\;10,\;15,\;21,\;?$ lists the triangular numbers. What number belongs in the blank?
    Answer: 28 When you encounter a sequence problem, your first step is identifying the pattern by examining how each term relates to the previous ones. Triangular numbers represent the sum of consecutive integers starting from 1. The first triangular number is $1$, the second is $1+2=3$, the third is $1+2+3=6$, and so on. You can also find the pattern by looking at the differences between consecutive terms: $3-1=2$, $6-3=3$, $10-6=4$, $15-10=5$, $21-15=6$. Notice that each difference increases by 1. Following this pattern, the next difference should be $7$, so the seventh triangular number is $21+7=28$. You can verify this using the formula for the $n$th triangular number: $\frac{n(n+1)}{2}$. For $n=7$: $\frac{7 \times 8}{2} = 28$. Looking at the wrong answers: Choice (A) 27 might tempt you if you mistakenly added 6 again instead of 7, giving $21+6=27$. Choice (C) 30 could result from incorrectly assuming the differences increase by 2 each time, leading to $21+9=30$. Choice (D) 36 might come from misapplying patterns or confusing triangular numbers with other sequences. The correct answer is (B) 28. For sequence problems on the TACHS, always identify the pattern in the differences between terms first. Many sequences follow predictable difference patterns, and recognizing these will help you solve similar problems quickly and accurately.
13. Which expression gives the $n$th term of the sequence $4,\;7,\;10,\;13,\;\dots$?
    Answer: $3n+1$ When you encounter a sequence problem, you're looking for a pattern that connects each term's position (n) to its value. This sequence increases by the same amount each time, making it an arithmetic sequence. First, identify the pattern. Looking at consecutive terms: 7 - 4 = 3, 10 - 7 = 3, 13 - 10 = 3. The sequence increases by 3 each time, so this is an arithmetic sequence with a common difference of 3. For arithmetic sequences, the general formula is: $a_n = a_1 + (n-1)d$, where $a_1$ is the first term and $d$ is the common difference. Here, $a_1 = 4$ and $d = 3$, so: $a_n = 4 + (n-1)(3) = 4 + 3n - 3 = 3n + 1$. You can verify this: when $n = 1$, $3(1) + 1 = 4$ ✓; when $n = 2$, $3(2) + 1 = 7$ ✓. Choice A) $2n + 2$ gives you 4, 6, 8, 10... - this increases by 2, not 3, and the second term would be 6, not 7. Choice C) $n^2 + 3$ gives you 4, 7, 12, 16... - while the first two terms match, this is quadratic growth, not linear, so the third term jumps to 12 instead of 10. Choice D) $4n$ gives you 4, 8, 12, 16... - this increases by 4 and misses every term except the first. Remember: for arithmetic sequences, always find the common difference first, then use the pattern $(\text{common difference}) \times n + \text{adjustment}$ to build your formula.

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