This question tests middle school pattern recognition skills: finding a missing term in a sequence. Pattern recognition involves identifying the consistent rule that governs the sequence, such as arithmetic difference or geometric ratio. In the given sequence 50, 45, __, 35, 30, each term decreases by 5, forming an arithmetic sequence with a common difference of -5. The correct answer, '40', fits the pattern because 45 - 5 = 40 and 40 - 5 = 35, maintaining the decrease. A common mistake is choosing '42', using a varying difference. To help students: Teach them to handle negative differences. Encourage working backward. Watch for: sign errors in subtraction.

When you encounter a sequence problem, your first step is to look for the pattern connecting 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: $$7-3=4$$, $$15-7=8$$, $$31-15=16$$, $$63-31=32$$. The differences are $$4, 8, 16, 32$$—each difference doubles! This means the next difference should be $$32 \times 2 = 64$$.

Therefore, the next term is $$63 + 64 = 127$$.

You can also spot another pattern: each term equals double the previous term plus 1. Check it: $$3 \times 2 + 1 = 7$$, $$7 \times 2 + 1 = 15$$, $$15 \times 2 + 1 = 31$$, $$31 \times 2 + 1 = 63$$. So the next term is $$63 \times 2 + 1 = 127$$.

Choice A ($$95$$) might tempt you if you incorrectly assumed the differences increase by 12 each time ($$4, 8, 16$$ could look like adding 4, then 8, then 12). Choice C ($$159$$) could result from thinking each term triples minus some constant. Choice D ($$191$$) might come from assuming the differences follow an arithmetic sequence with a larger common difference.

The key strategy for sequence problems is to always check the differences between terms first, and if those don't follow a simple pattern, look at the differences between the differences. Many SSAT sequences involve exponential patterns like doubling, so keep that in mind when the first differences form a geometric sequence.

When you encounter a sequence problem, your first step is identifying the pattern by examining how each term relates to the previous one. Look at the differences between consecutive terms, and if those differences don't form an obvious pattern, check if there's a multiplicative relationship or a more complex rule.

Let's find the pattern in this sequence: $$5, 11, 23, 47$$. The differences between consecutive terms are: $$11-5=6$$, $$23-11=12$$, and $$47-23=24$$. Notice that these differences ($$6, 12, 24$$) are doubling each time! This means each term follows the rule: next term = current term + (previous difference × 2).

Following this pattern: The 5th term is $$47 + 24 \times 2 = 47 + 48 = 95$$. The 6th term is $$95 + 48 \times 2 = 95 + 96 = 191$$.

Looking at the wrong answers: Choice C gives $$95$$, which is actually the 5th term—a common error when you lose track of which term you're finding. Choice D gives $$189$$, likely from a calculation error like $$95 + 94 = 189$$, possibly confusing the doubling pattern. Choice B gives $$383$$, which might result from incorrectly thinking the terms themselves double ($$47 \times 2 = 94$$, then continuing incorrectly).

The correct answer is A) $$191$$.

Strategy tip: In sequence problems, always write out the differences between terms first. If the first differences don't show a clear pattern, check if the differences themselves follow a pattern—this "second difference" approach catches many sequence types on the SSAT.

When you encounter an arithmetic sequence, you're working with a pattern where each term increases by the same amount (called the common difference). Here, the sequence $$8, 13, 18, 23, ...$$ increases by $$5$$ each time, so the common difference is $$d = 5$$.

To find which term equals $$73$$, use the arithmetic sequence formula: $$a_n = a_1 + (n-1)d$$, where $$a_n$$ is the $$n$$th term, $$a_1$$ is the first term, and $$d$$ is the common difference.

Substituting what we know: $$73 = 8 + (n-1)(5)$$. Solving for $$n$$: $$73 = 8 + 5n - 5$$ $$73 = 3 + 5n$$ $$70 = 5n$$ $$n = 14$$

So $$73$$ is the 14th term, making B correct.

Let's check why the other answers are wrong by calculating what those terms would actually be:

• A) The 13th term: $$a_{13} = 8 + (13-1)(5) = 8 + 60 = 68$$
• C) The 15th term: $$a_{15} = 8 + (15-1)(5) = 8 + 70 = 78$$
• D) The 16th term: $$a_{16} = 8 + (16-1)(5) = 8 + 75 = 83$$

These distractors likely represent common calculation errors: choice A might result from miscounting or arithmetic mistakes, while C and D come from adding too many steps.

Strategy tip: Always double-check arithmetic sequence problems by plugging your answer back into the formula. This catches calculation errors and confirms you've found the right term position.

This sequence is called the Fibonacci sequence, where each term equals the sum of the two terms before it. When you encounter a sequence problem like this, your approach should be to identify the pattern and then carefully calculate term by term until you reach the target.

Starting with the given terms $$1, 1, 2, 3, 5, 8, 13$$, you can find the remaining terms by adding consecutive pairs:

• 8th term: $$8 + 13 = 21$$
• 9th term: $$13 + 21 = 34$$
• 10th term: $$21 + 34 = 55$$

Therefore, the 10th term is $$55$$, which is choice B.

Let's examine why the other answers are incorrect. Choice A ($$34$$) is actually the 9th term in the sequence—this represents a common error where students miscount their position or stop one term too early. Choice C ($$89$$) would be the 11th term if you continued the sequence ($$34 + 55 = 89$$), showing what happens when you go one step too far. Choice D ($$144$$) would be the 12th term ($$55 + 89 = 144$$), representing an even larger counting error.

The key strategy for sequence problems is to write out each term methodically and keep careful track of your position. Don't try to skip steps or use shortcuts—the sequential nature means one error will throw off everything that follows. Always double-check by counting your terms to ensure you've reached exactly the position requested.

When you encounter a sequence where the pattern isn't immediately obvious, examine the differences between consecutive terms to find a hidden pattern.

Let's find the differences between consecutive terms in the sequence $$3, 5, 9, 15, 23, ...$$:

• $$5 - 3 = 2$$
• $$9 - 5 = 4$$
• $$15 - 9 = 6$$
• $$23 - 15 = 8$$

The differences form the sequence $$2, 4, 6, 8, ...$$ - consecutive even numbers! This means each difference increases by 2.

Continuing this pattern:

• 5th term: $$23 + 10 = 33$$ (since the next difference is 10)
• 6th term: $$33 + 12 = 45$$ (since the next difference is 12)
• 7th term: $$45 + 14 = 59$$ (since the next difference is 14)

Wait - let me recalculate the 7th term. The 6th term is 45, so the 7th term is $$45 + 14 = 59$$. However, this gives us answer C, not A.

Let me verify: if the 6th term is 45, then A) $$45$$ represents the 6th term, not the 7th. The 7th term would be C) $$59$$.

Looking at the wrong answers: B) $$51$$ might result from adding 6 instead of 14 to 45. D) $$69$$ could come from miscounting terms or adding incorrectly.

The key strategy here is recognizing second-order patterns - when the original sequence isn't arithmetic or geometric, check if the differences between terms follow a pattern. This technique appears frequently on standardized tests and helps solve many "mystery sequences."

When you encounter a number sequence problem, your goal is to identify the pattern or rule that generates each term from the previous ones.

Let's examine how each term relates to the one before it:

• From 2 to 5: $$2 \times 2 + 1 = 5$$
• From 5 to 11: $$5 \times 2 + 1 = 11$$
• From 11 to 23: $$11 \times 2 + 1 = 23$$
• From 23 to 47: $$23 \times 2 + 1 = 47$$

The pattern is clear: multiply each term by 2, then add 1 to get the next term. Applying this rule to find the term after 47: $$47 \times 2 + 1 = 94 + 1 = 95$$. This confirms that B) $$95$$ is correct.

Let's see why the other answers miss the mark. Choice A) $$71$$ would result from adding 24 to 47, but this doesn't follow the consistent doubling-plus-one pattern we identified. Choice C) $$119$$ is what you'd get if you used the rule $$47 \times 2 + 25$$, but there's no justification for adding 25 instead of 1. Choice D) $$143$$ would come from tripling 47 and adding 2, which completely abandons the established pattern.

For sequence problems on the SSAT, always test your suspected pattern against multiple terms to confirm it works consistently. Don't assume the pattern is simply adding or subtracting a constant—look for multiplication, division, or combination rules like the "double and add one" pattern here.

When you encounter a sequence problem, your first step is to identify the pattern by examining how the terms relate to each other. Let's look at the differences between consecutive terms: $$6-2=4$$, $$12-6=6$$, $$20-12=8$$, $$30-20=10$$. The differences are $$4, 6, 8, 10$$, which increase by $$2$$ each time.

This tells us we have a second-order pattern. Looking more closely, each term can be written as $$n(n+1)$$ where $$n$$ is the term number: the 1st term is $$1 \times 2 = 2$$, the 2nd term is $$2 \times 3 = 6$$, the 3rd term is $$3 \times 4 = 12$$, and so on. Therefore, the 8th term is $$8 \times 9 = 72$$.

Answer choice A ($$56$$) represents $$7 \times 8$$, which would be the 7th term, not the 8th. This is a common error when students miscount the position. Answer choice C ($$90$$) equals $$9 \times 10$$, which would be the 9th term—another position-counting mistake. Answer choice D ($$110$$) doesn't follow the pattern at all and might represent a calculation error or incorrect formula application.

The correct answer is B ($$72$$).

Strategy tip: For sequence problems, always look for patterns in the differences between terms first. If the first differences aren't constant, check if the second differences are constant—this often reveals quadratic patterns like $$n(n+1)$$ that appear frequently on standardized tests.

When you see a sequence like 1, 3, 6, 10, 15, 21, recognize these as triangular numbers - each represents the sum of consecutive integers starting from 1. The pattern shows: 1 = 1, 3 = 1+2, 6 = 1+2+3, 10 = 1+2+3+4, and so on.

To find which sum equals 120, you can use the formula for the sum of consecutive integers from 1 to n: $$\frac{n(n+1)}{2} = 120$$

Multiplying both sides by 2: $$n(n+1) = 240$$

You need two consecutive numbers whose product is 240. Testing values around the square root of 240 (about 15.5):

• 15 × 16 = 240 ✓

So n = 15, meaning 1 + 2 + 3 + ... + 15 = 120.

Choice A (1 + 2 + 3 + ... + 15) is correct because $$\frac{15 \times 16}{2} = 120$$.

Choice B (up to 16) gives $$\frac{16 \times 17}{2} = 136$$, which is too large.

Choice C (up to 17) gives $$\frac{17 \times 18}{2} = 153$$, even larger.

Choice D (up to 18) gives $$\frac{18 \times 19}{2} = 171$$, the largest.

Strategy tip: For triangular number problems, memorize the formula $$\frac{n(n+1)}{2}$$ for the sum 1 + 2 + ... + n. When working backwards from a given sum, set up the equation and look for consecutive integers whose product equals twice your target sum. This saves time compared to adding each sequence manually.

When you encounter a sequence pattern problem, your goal is to identify the rule that transforms each term into the next one. Start by examining the differences between consecutive terms to find the pattern.

Let's analyze the given sequence: $$7, 10, 16, 28, 52, ...$$

First, look at the differences between consecutive terms:

• $$10 - 7 = 3$$
• $$16 - 10 = 6$$
• $$28 - 16 = 12$$
• $$52 - 28 = 24$$

Notice that the differences are $$3, 6, 12, 24$$. Each difference doubles the previous one! This means the rule is: to get the next term, add double the previous difference.

Following this pattern, the next difference should be $$24 \times 2 = 48$$. Therefore, the next term is $$52 + 48 = 100$$.

Looking at the wrong answers: Choice (A) $$76$$ would result from adding $$24$$ again instead of doubling it to $$48$$. Choice (C) $$104$$ might come from miscalculating the doubling pattern or arithmetic errors. Choice (D) $$108$$ could result from incorrectly identifying the pattern as adding a fixed amount or making computational mistakes.

The correct answer is (B) $$100$$.

Study tip: For sequence problems, always calculate the differences between consecutive terms first. If those differences don't form an obvious pattern, look at the ratios between consecutive terms, or check if the differences themselves follow a pattern (like doubling, as in this problem). This systematic approach will help you crack most sequence patterns on the SSAT.