1.4 Sequences, Series, and Number Properties

1.4 Sequences, Series, and Number Properties

The Math Skills Test (MST) explicitly samples sequences and properties of numbers alongside arithmetic and algebra. These items reward pattern recognition and a handful of memorized rules far more than heavy computation, which matters because the ASTB-E is taken without a calculator under tight per-item time pressure. Being able to name a pattern in a few seconds and apply the right formula is what separates a fast, accurate examinee from one who runs out of time.

Arithmetic Sequences

An arithmetic sequence is an ordered list of numbers in which each term is obtained by adding a fixed value, the common difference (d), to the previous term. In the sequence 4, 7, 10, 13, 16, the common difference is d = 3.

Two formulas cover almost every arithmetic-sequence item:

• nth term: a_n = a_1 + (n - 1)d
• Sum of the first n terms (arithmetic series): S_n = n/2 × (a_1 + a_n)

Worked example. Find the 20th term of 4, 7, 10, 13, …

Here a_1 = 4 and d = 3, so a_20 = 4 + (20 - 1)(3) = 4 + 57 = 61. To sum those 20 terms, S_20 = 20/2 × (4 + 61) = 10 × 65 = 650.

Geometric Sequences

A geometric sequence multiplies each term by a fixed common ratio (r) instead of adding. In 3, 6, 12, 24, 48, the ratio is r = 2. Geometric growth accelerates (or decays) far faster than arithmetic growth — a favorite MST trap is to assume a sequence is arithmetic when the gaps are actually multiplying.

• nth term: a_n = a_1 × r^(n-1)

Worked example. Find the 6th term of 5, 10, 20, 40, …

With a_1 = 5 and r = 2, a_6 = 5 × 2^(6-1) = 5 × 32 = 160.

Recognizing the Pattern Fast

Use this quick decision order on any "what comes next" item:

1. Constant difference? Subtract consecutive terms. If the gap is the same, it is arithmetic.
2. Constant ratio? Divide consecutive terms. If the quotient is the same, it is geometric.
3. Otherwise test squares (1, 4, 9, 16), cubes (1, 8, 27), Fibonacci-style addition (each term is the sum of the two before it), or an alternating rule (two interleaved patterns).

Fibonacci-style example. In 2, 3, 5, 8, 13, …, no single difference or ratio fits, but each term is the sum of the two before it (5 = 2 + 3, 8 = 3 + 5). The next term is 8 + 13 = 21.

Alternating example. In 1, 10, 2, 20, 3, 30, …, two patterns are interleaved: the odd positions count 1, 2, 3, … while the even positions count 10, 20, 30, …. The next term (an odd position) is 4. Whenever gaps look erratic, check whether two separate sequences are woven together.

Difference-of-differences. For 2, 5, 10, 17, 26, …, the first differences are 3, 5, 7, 9 — not constant, but they rise by 2 each time. That signals a perfect-square-plus-one pattern: 1²+1, 2²+1, 3²+1, …, so the next term is 6²+1 = 37. Taking a second round of differences is the standard trick when the first differences themselves form a simple pattern.

Number Properties

Many MST items hinge on properties of numbers rather than sequences:

• Prime vs. composite: A prime has exactly two factors, 1 and itself (2, 3, 5, 7, 11, …); 2 is the only even prime. A composite has more than two factors.
• Factors and multiples: The greatest common factor (GCF) is the largest number dividing two values; the least common multiple (LCM) is the smallest number both divide into. GCF(12, 18) = 6; LCM(12, 18) = 36.
• Divisibility shortcuts: divisible by 2 if even; by 3 if the digit sum is divisible by 3; by 5 if it ends in 0 or 5; by 9 if the digit sum is divisible by 9.
• Integer sign and parity rules: even ± even = even; odd + odd = even; even × any = even; a negative times a negative is positive.

Quick-Reference Table

Because these items are computation-light, they are among the fastest points on the MST — bank the time you save here for the heavier word problems later in the subtest.