When you encounter a sequence where each term is found by multiplying the previous term by the same number, you're working with a geometric sequence. The key is identifying the pattern and using it systematically.

This sequence starts with 2 and multiplies by 3 each time: 2, 6, 18, 54, 162, ... To find the sixth term, continue the pattern: $$162 \times 3 = 486$$. So the first six terms are: 2, 6, 18, 54, 162, 486.

Now sum them: $$2 + 6 + 18 + 54 + 162 + 486 = 728$$. This confirms answer choice (A) is correct.

Looking at the wrong answers: (B) 486 is simply the sixth term itself, not the sum of all six terms. This is a common trap where students find the correct final term but forget to add up all the terms. (C) 972 is exactly $$728 + 244$$, which might result from accidentally including a seventh term or making an arithmetic error in addition. (D) 364 is exactly half of 728, suggesting a student might have made a calculation error or somehow only counted half the terms.

For geometric sequence problems on the ISEE, always write out each term clearly rather than trying to do the calculations in your head. The numbers get large quickly when multiplying, and it's easy to make arithmetic mistakes. Also, pay careful attention to what the question asks for—individual terms versus sums are common mix-ups in answer choices.

When you encounter a pattern problem involving geometric arrangements, look for the underlying mathematical relationship between the position and the number of elements.

Let's examine this dot pattern carefully. The first square has 1 dot, the second has 4 dots, the third has 9 dots, and the fourth has 16 dots. Notice that these numbers are perfect squares: $$1^2 = 1$$, $$2^2 = 4$$, $$3^2 = 9$$, and $$4^2 = 16$$. This makes sense because dots arranged in a square formation would have equal rows and columns—a 3×3 arrangement gives 9 dots, a 4×4 arrangement gives 16 dots, and so on.

Following this pattern, the nth square contains $$n^2$$ dots. Therefore, the 15th square contains $$15^2 = 225$$ dots.

Let's examine why the other answers are wrong. Answer B (240) might tempt you if you mistakenly think the pattern increases by 15 each time, but that's not how this sequence works. Answer C (210) could result from incorrectly calculating $$15 \times 14$$, perhaps thinking you need to multiply consecutive numbers. Answer D (256) equals $$16^2$$, which would be the 16th square, not the 15th—a common off-by-one error.

Study tip: When you see geometric patterns on the ISEE, always check if the numbers are perfect squares, cubes, or other powers. Pattern recognition combined with understanding the geometric structure (like square arrangements) will help you identify the rule quickly and avoid calculation traps.

When you encounter an arithmetic sequence problem asking for the sum of terms, you need to identify the pattern and apply the arithmetic series formula.

First, let's confirm this is arithmetic by checking that consecutive terms have the same difference: $$8-5=3$$, $$11-8=3$$, $$14-11=3$$. The common difference is $$d=3$$ and the first term is $$a_1=5$$.

To find the sum of the first 20 terms, use the arithmetic series formula: $$S_n = \frac{n}{2}(2a_1 + (n-1)d)$$, where $$n$$ is the number of terms.

Substituting our values: $$S_{20} = \frac{20}{2}(2(5) + (20-1)(3)) = 10(10 + 19 \cdot 3) = 10(10 + 57) = 10(67) = 670$$

Choice A (670) is correct as shown above.

Choice B (650) likely results from a calculation error, perhaps using $$d=2$$ instead of $$d=3$$, or making an arithmetic mistake when computing $$19 \times 3$$.

Choice C (630) could come from incorrectly using $$n=19$$ instead of $$n=20$$, or from other computational errors in the formula application.

Choice D (690) might result from using the wrong formula or making an error like adding instead of multiplying somewhere in the calculation.

Strategy tip: Always double-check that you've correctly identified the common difference by testing multiple consecutive pairs. Then be extra careful with arithmetic when substituting into the formula—small calculation errors are the most common trap in series problems.

When you encounter a sequence problem, your goal is to identify the underlying pattern by examining how the terms relate to their position numbers. Start by testing each given formula against the actual sequence values.

Let's check each option systematically. For $$a_n = n^2 + 1$$: when $$n = 1$$, we get $$1^2 + 1 = 2$$ ✓; when $$n = 2$$, we get $$2^2 + 1 = 5$$ ✓; when $$n = 3$$, we get $$3^2 + 1 = 10$$ ✓; when $$n = 4$$, we get $$4^2 + 1 = 17$$ ✓; when $$n = 5$$, we get $$5^2 + 1 = 26$$ ✓. This formula works perfectly for all given terms.

Option B ($$a_n = 3n - 1$$) gives us: $$3(1) - 1 = 2$$ ✓, but $$3(2) - 1 = 5$$ ✓, then $$3(3) - 1 = 8$$ ✗ (should be 10). This linear formula fails at the third term.

Option C ($$a_n = 2n + 3$$) produces: $$2(1) + 3 = 5$$ ✗ (should be 2). This fails immediately at the first term.

Option D ($$a_n = n^2 + n$$) yields: $$1^2 + 1 = 2$$ ✓, but $$2^2 + 2 = 6$$ ✗ (should be 5). This fails at the second term.

The correct answer is A.

Study tip: When analyzing sequences, always test formulas against multiple terms, not just the first one or two. Quadratic sequences (involving $$n^2$$) are common on standardized tests, so if you notice the differences between consecutive terms aren't constant, consider quadratic patterns first.

Sequence questions test your ability to recognize patterns and continue them systematically. The Fibonacci sequence is a special pattern where each term equals the sum of the two terms immediately before it.

Starting with the given terms 1, 1, 2, 3, 5, 8, 13, you need to continue this pattern to find the 10th term. Let's work systematically:

• Term 1: 1
• Term 2: 1
• Term 3: 1 + 1 = 2
• Term 4: 1 + 2 = 3
• Term 5: 2 + 3 = 5
• Term 6: 3 + 5 = 8
• Term 7: 5 + 8 = 13
• Term 8: 8 + 13 = 21
• Term 9: 13 + 21 = 34
• Term 10: 21 + 34 = 55

The 10th term is 55, making (A) correct.

Looking at the wrong answers: (B) 89 would actually be the 11th term (34 + 55), representing a common off-by-one counting error. (C) 34 is the 9th term, showing you stopped one step too early. (D) 21 is the 8th term, indicating you stopped two steps short of the target.

When working with sequences, always write out each step rather than trying to jump ahead mentally. This prevents counting errors and ensures you apply the pattern correctly. Double-check by counting your terms carefully—sequence problems often include answer choices that correspond to nearby terms to catch careless mistakes.

When you encounter a recursive sequence, you're working with a pattern where each term depends on previous terms. This particular sequence follows the same rule as the famous Fibonacci sequence: each term equals the sum of the two preceding terms.

Starting with the given values $$a_1 = 2$$ and $$a_2 = 3$$, you can build the sequence step by step using the rule $$a_n = a_{n-1} + a_{n-2}$$:

$$a_3 = a_2 + a_1 = 3 + 2 = 5$$

$$a_4 = a_3 + a_2 = 5 + 3 = 8$$

$$a_5 = a_4 + a_3 = 8 + 5 = 13$$

$$a_6 = a_5 + a_4 = 13 + 8 = 21$$

Therefore, $$a_6 = 21$$, making (C) correct.

Looking at the wrong answers: (A) 18 likely comes from miscalculating one of the middle terms and carrying that error forward. (B) 13 is actually $$a_5$$, suggesting someone stopped calculating one step too early or confused which term the question was asking for. (D) 8 equals $$a_4$$, indicating an even earlier miscounting of terms.

The key strategy for recursive sequences is to work methodically, one term at a time, and double-check your arithmetic at each step since errors compound. Also, pay careful attention to which term number the question asks for—it's easy to lose track when calculating several terms in sequence. Write out each step clearly to avoid careless mistakes.

Geometric sequences follow a pattern where each term is found by multiplying the previous term by a constant ratio. When you see a geometric sequence problem with non-consecutive terms given, you need to use the general formula: $$a_n = a_1 \cdot r^{n-1}$$, where $$a_1$$ is the first term and $$r$$ is the common ratio.

Since the second term is 12 and the fifth term is 96, you can write two equations: $$a_2 = a_1 \cdot r = 12$$ and $$a_5 = a_1 \cdot r^4 = 96$$. To find the common ratio, divide the fifth term equation by the second term equation: $$\frac{a_1 \cdot r^4}{a_1 \cdot r} = \frac{96}{12}$$. This simplifies to $$r^3 = 8$$, so $$r = 2$$.

Now substitute back into the second term equation: $$a_1 \cdot 2 = 12$$, which gives you $$a_1 = 6$$.

Looking at the wrong answers: B) 8 would give you a second term of 16 (not 12) when multiplied by the ratio of 2. C) 4 would give you a second term of 8, which is incorrect. D) 3 would give you a second term of 6, also incorrect.

You can verify: starting with 6, the sequence is 6, 12, 24, 48, 96, confirming that A) 6 is correct.

Strategy tip: For geometric sequence problems, always look for ways to eliminate the unknown first term by creating ratios between the given terms—this often reveals the common ratio quickly.

When you encounter triangular number sequences, you're working with a specific pattern where each term represents the sum of consecutive integers starting from 1. The formula $$T_n = \frac{n(n+1)}{2}$$ gives you any triangular number directly without having to add up all the preceding terms.

To find the difference between the 8th and 6th triangular numbers, calculate each term using the formula. For the 8th triangular number: $$T_8 = \frac{8(8+1)}{2} = \frac{8 \times 9}{2} = \frac{72}{2} = 36$$. For the 6th triangular number: $$T_6 = \frac{6(6+1)}{2} = \frac{6 \times 7}{2} = \frac{42}{2} = 21$$. The difference is $$36 - 21 = 15$$.

Looking at the wrong answers: B) 13 might result from calculation errors in the formula or confusing which terms to subtract. C) 14 could come from mistakenly finding the difference between consecutive triangular numbers (like $$T_7 - T_6$$) rather than the specific terms asked. D) 16 likely stems from arithmetic mistakes when applying the formula or incorrectly calculating $$T_8$$.

Remember that triangular number problems often test your ability to use the given formula accurately rather than recognizing complex patterns. Always substitute carefully into $$T_n = \frac{n(n+1)}{2}$$, double-check your arithmetic, and make sure you're finding the difference between the correct terms specified in the question.

When you encounter arithmetic sequence problems, remember that these sequences follow a predictable pattern where each term differs from the previous by a constant amount called the common difference.

To find any term in an arithmetic sequence, use the formula: $$a_n = a_1 + (n-1)d$$, where $$a_n$$ is the nth term, $$a_1$$ is the first term, and $$d$$ is the common difference.

Since you know the 6th term is 23 and the common difference is -4, first find the first term. Using $$a_6 = a_1 + (6-1)(-4)$$: $$23 = a_1 + 5(-4)$$, so $$23 = a_1 - 20$$, which gives $$a_1 = 43$$.

Now find the 12th term: $$a_{12} = 43 + (12-1)(-4) = 43 + 11(-4) = 43 - 44 = -1$$.

Looking at the wrong answers: B) 1 might result from a sign error in your calculations or forgetting that the common difference is negative. C) -5 could come from miscounting the number of steps between terms or making an arithmetic mistake. D) 3 likely results from multiple computational errors, possibly confusing addition and subtraction.

The key strategy for arithmetic sequences is to always establish what you know first, then systematically work toward what you need to find. Don't try to jump directly between distant terms—use the formula methodically. Also, pay careful attention to negative common differences, as they create decreasing sequences where later terms are smaller than earlier ones.

When you encounter a sequence problem, your first step is to identify the pattern by examining how each term relates to its position in the sequence.

Looking at this sequence: $$\frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \frac{4}{5}, \frac{5}{6}, ...$$

Notice that in each fraction, the numerator is one less than the denominator. More specifically, if we call the position in the sequence $$n$$, then:

• 1st term: $$\frac{1}{2} = \frac{1}{1+1}$$
• 2nd term: $$\frac{2}{3} = \frac{2}{2+1}$$
• 3rd term: $$\frac{3}{4} = \frac{3}{3+1}$$

The pattern is clear: the $$n$$th term equals $$\frac{n}{n+1}$$

For the 20th term, substitute $$n = 20$$: $$\frac{20}{20+1} = \frac{20}{21}$$

Looking at the wrong answers: Choice B ($$\frac{19}{20}$$) represents the 19th term—a common error when students miscount positions. Choice C ($$\frac{21}{22}$$) would be the 21st term, another off-by-one mistake. Choice D ($$\frac{20}{19}$$) flips the pattern by putting the larger number in the numerator, which breaks the sequence's structure where each term is less than 1.

The answer is A: $$\frac{20}{21}$$

Study tip: For sequence problems, always write out the first few terms with their position numbers, identify the pattern algebraically, then substitute carefully. Double-check by verifying your formula works for the given terms before applying it to find the unknown term.