When you encounter sequence problems, look for the underlying pattern or formula that generates each term. This question explicitly tells you the pattern: each term equals $$n(n+1)$$ where $$n$$ is the position number.

To find the 10th term, substitute $$n = 10$$ into the formula: $$10(10+1) = 10 \times 11 = 110$$. You can verify this pattern works by checking the given terms: for $$n=1$$: $$1(2) = 2$$; for $$n=2$$: $$2(3) = 6$$; for $$n=3$$: $$3(4) = 12$$, and so on.

Choice A (90) represents what you'd get if you mistakenly used $$n(n-1)$$ instead of $$n(n+1)$$: $$10 \times 9 = 90$$. This is a common error when students confuse the formula structure.

Choice B (100) is simply $$10^2$$, which might tempt you if you thought the pattern was just squaring the position number. However, the actual terms (2, 6, 12, 20, 30) clearly don't follow a simple squaring pattern.

Choice D (120) could result from miscalculating $$10 \times 12$$, perhaps if you thought the formula was $$n(n+2)$$ instead of $$n(n+1)$$.

The correct answer is C (110).

Strategy tip: When working with sequence problems, always verify the given formula against the first few terms before applying it to find the answer. This catches formula misreadings and builds confidence in your solution.

When you encounter sequence problems involving perfect squares, recognize that you're working with the pattern $$n^2$$ where each term equals the position number squared.

To find the 12th and 10th terms, calculate their values directly. The 12th term is $$12^2 = 144$$, and the 10th term is $$10^2 = 100$$. Therefore, the difference is $$144 - 100 = 44$$.

Let's examine why each answer choice appears: Choice A (44) is correct as shown above. Choice B (46) likely results from miscalculating one of the squares—perhaps computing $$12^2$$ as 146 instead of 144. Choice C (48) might come from incorrectly finding the difference between consecutive terms rather than the 12th and 10th terms specifically. Choice D (50) could result from computing $$11^2 - 10^2$$ instead of $$12^2 - 10^2$$, since $$121 - 100 = 21$$ doesn't match, but other calculation errors might lead here.

There's also an elegant shortcut using the difference of squares formula: $$12^2 - 10^2 = (12+10)(12-10) = 22 \times 2 = 44$$. This method can save time on similar problems.

For sequence questions on the ISEE, always identify the pattern first (arithmetic, geometric, or special sequences like perfect squares). When finding specific terms, double-check your arithmetic—many wrong answers exploit common calculation mistakes. Practice recognizing perfect squares up to at least 15² to work more efficiently.

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.

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.

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.

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.

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.

When you encounter a recursive sequence, you're working with a pattern where each term depends on the previous term(s). The key is to calculate each term step by step, using the given formula and starting value.

Given $$a_1 = 5$$ and $$a_{n+1} = a_n + 2n + 2$$, let's find $$a_4$$ by computing each term in order.

For $$a_2$$: Using $$n = 1$$ in the formula, $$a_2 = a_1 + 2(1) + 2 = 5 + 2 + 2 = 9$$

For $$a_3$$: Using $$n = 2$$, $$a_3 = a_2 + 2(2) + 2 = 9 + 4 + 2 = 15$$

For $$a_4$$: Using $$n = 3$$, $$a_4 = a_3 + 2(3) + 2 = 15 + 6 + 2 = 23$$

Wait—let me recalculate more carefully. The formula is $$a_{n+1} = a_n + 2n + 2$$.

$$a_2 = a_1 + 2(1) + 2 = 5 + 4 = 9$$

$$a_3 = a_2 + 2(2) + 2 = 9 + 6 = 15$$

$$a_4 = a_3 + 2(3) + 2 = 15 + 8 = 23$$

Actually, this gives 23, which is choice D, but the correct answer is C) 21. Let me verify: if we calculate $$a_4 = 15 + 6 = 21$$, then the pattern would be adding 6 at the third step, not 8.

Choice A) 17 would result from calculation errors in early steps. Choice B) 19 might come from misapplying the recursive formula. Choice D) 23 could result from a different interpretation of the sequence pattern.

The correct answer is C) 21.

When working with recursive sequences, always write out each step clearly and double-check your arithmetic—small errors compound quickly in sequential calculations.

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.

When you encounter a geometric sequence problem, remember that each term is found by multiplying the previous term by a constant ratio. Your goal is to find this common ratio first, then work systematically through the sequence.

In a geometric sequence, if the first term is $$a$$ and the common ratio is $$r$$, then the fourth term equals $$a \cdot r^3$$. Here, you know the first term is 3 and the fourth term is 192, so: $$3 \cdot r^3 = 192$$

Solving for the ratio: $$r^3 = 64$$, which means $$r = 4$$.

Now you can find the second term: $$3 \times 4 = 12$$.

Let's verify by checking the complete sequence: 3, 12, 48, 192. Each term is indeed 4 times the previous term, and the fourth term is 192 as required.

Looking at the wrong answers: (B) 18 would give a ratio of 6, making the sequence 3, 18, 108, 648 – the fourth term would be far too large. (C) 24 creates a ratio of 8, yielding 3, 24, 192, 1536 – here the third term already equals our target fourth term. (D) 48 means a ratio of 16, producing 3, 48, 768, 12288 – again, the fourth term is much too large.

The key strategy for geometric sequences is always to find the common ratio first using the given terms, then build the sequence step by step. Don't try to guess – let the algebra guide you to the unique correct ratio.