4.5 Patterns and Sequences

Why Patterns and Sequences Matter on the SHSAT

Pattern-recognition questions ask you to identify the rule governing a list of numbers and then use that rule to find a later term or a sum. On the SHSAT, these questions appear both as standalone items ("What is the 10th term of the sequence 3, 7, 11, ...?") and embedded in word problems ("A worker's weekly raise follows a pattern..."). Sequences also underpin number-pattern problems, function-rate-of-change problems, and geometric similarity scaling. Mastering two basic sequence types — arithmetic and geometric — unlocks the vast majority of these items.

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Arithmetic Sequences

An arithmetic sequence is a list of numbers in which the difference between consecutive terms is constant. This constant difference is called the common difference, denoted d.

General form: a_1, a_1 + d, a_1 + 2d, a_1 + 3d, ...

nth Term Formula

The nth term of an arithmetic sequence is:

$a_n = a_1 + (n - 1) d$

Example: Find the 20th term of 5, 8, 11, 14, ... Here a_1 = 5 and d = 3. Using the formula: a_20 = 5 + (20 - 1)(3) = 5 + 57 = 62.

Sum of the First n Terms

The sum of the first n terms is:

$S_n = \frac{n}{2} (a_1 + a_n)$

Example: Find the sum of the first 20 terms of the sequence above. S_20 = (20 / 2)(5 + 62) = 10 * 67 = 670.

Word-Problem Cues

Phrases such as "each week the value increases by the same amount," "steady weekly raise," or "constant rate of increase" signal an arithmetic sequence.

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Geometric Sequences

A geometric sequence is a list of numbers in which the ratio between consecutive terms is constant. This constant ratio is called the common ratio, denoted r.

General form: a_1, a_1r, a_1r^2, a_1*r^3, ...

nth Term Formula

The nth term of a geometric sequence is:

$a_n = a_1 \cdot r^{n-1}$

Example: Find the 7th term of 2, 6, 18, 54, ... Here a_1 = 2 and r = 3. Using the formula: a_7 = 2 * 3^(7-1) = 2 * 3^6 = 2 * 729 = 1458.

Common Ratio from Two Terms

If you know two non-adjacent terms, divide to find r: Example: If the 2nd term is 12 and the 5th term is 324, then a_2 = a_1r and a_5 = a_1r^4, so a_5 / a_2 = r^3 = 324 / 12 = 27, giving r = 3.

Word-Problem Cues

Phrases such as "doubles every hour," "triples each year," or "grows by the same factor" signal a geometric sequence. Bacterial growth, compound interest, and bouncing-distance problems are classic SHSAT geometric-sequence contexts.

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Sum of a Finite Geometric Series

The sum of the first n terms of a geometric sequence is:

$S_n = a_1 \cdot \frac{1 - r^n}{1 - r}, \quad r \neq 1$

Example: Find the sum of the first 5 terms of 3, 6, 12, 24, ... S_5 = 3 * (1 - 2^5) / (1 - 2) = 3 * (1 - 32) / (-1) = 3 * 31 = 93.

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Other Repeating Patterns

Not every SHSAT pattern fits arithmetic or geometric form. Be ready for:

Repeating Patterns

A sequence like 4, 1, 7, 4, 1, 7, 4, 1, 7, ... repeats every 3 terms. To find the 50th term, divide 50 by 3: 50 = 16*3 + 2, so the 50th term is the same as the 2nd term of the cycle, which is 1.

Recursive Patterns (Fibonacci-style)

Each term depends on previous terms, e.g., 1, 1, 2, 3, 5, 8, 13, ... where each term is the sum of the two preceding terms. List terms explicitly until you reach the requested one — do not try to memorize a closed form.

Alternating-Sign Patterns

A sequence like 2, -2, 2, -2, ... has a closed form a_n = 2(-1)^(n+1). Watch for sign flips when computing a specific term.

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Common SHSAT Traps

• Off-by-one in the index: The first term corresponds to n = 1, not n = 0. The 5th term uses exponent 4 in a geometric sequence, not 5. Always substitute (n - 1) for the offset.
• Confusing arithmetic with geometric: A "doubling" sequence is geometric, not arithmetic. Adding the same amount is arithmetic; multiplying by the same factor is geometric.
• Mixing term value with term sum: Read carefully whether the question asks for the nth term or the sum of the first n terms. The two formulas differ.
• Sign errors with negative common ratios: If r = -2, the terms alternate sign. Compute at least three terms to confirm the pattern.

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Worked SHSAT-Style Example

A basketball player's daily free-throw count follows a pattern. On day 1 she makes 20 free throws. Each subsequent day she makes 4 more than the previous day. How many total free throws does she make in the first 12 days?

Identify the sequence: arithmetic with a_1 = 20 and d = 4. Find a_12: a_12 = 20 + (12 - 1)(4) = 20 + 44 = 64. Compute the sum: S_12 = (12 / 2)(20 + 64) = 6 * 84 = 504 free throws.