Mathfolis

Change in Arithmetic and Geometric Sequences

Unit 2 · Exponential and Logarithmic Functions

What AP Precalc asks here

Sequences are ordered lists of numbers. The two foundational families: arithmetic sequences, where each term is the previous plus a common difference dd; and geometric sequences, where each term is the previous times a common ratio rr. Arithmetic sequences are the discrete analog of linear functions; geometric sequences are the discrete analog of exponential functions. The College Board tests identification, computing a specific term, and recovering the parameters from two known terms.

Arithmetic sequence

Recursive
an=an−1+da_n = a_{n-1} + d
Explicit
an=a1+(n−1)da_n = a_1 + (n - 1) d

Geometric sequence

Recursive
an=r⋅an−1a_n = r \cdot a_{n-1}
Explicit
an=a1⋅rn−1a_n = a_1 \cdot r^{n - 1}
Caution: The exponent is (n−1)(n - 1), not nn. Term a1a_1 corresponds to zero steps from the start.
Type 1

Identify sequence type

Compute successive differences. If constant, the sequence is arithmetic. Otherwise, compute successive ratios. If those are constant, it's geometric.

Example 1
Identify the type of sequence with a1=5,a2=15,a3=45,a4=135,a5=405a_1 = 5, a_2 = 15, a_3 = 45, a_4 = 135, a_5 = 405. (A) Arithmetic with d=10d = 10 (B) Geometric with r=3r = 3 (C) Arithmetic with d=30d = 30 (D) Neither

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Type 2

Find the n-th term

Apply the explicit formula. Watch for the (n−1)(n - 1) exponent — easy to off-by-one.

Example 2
For an arithmetic sequence with a1=7a_1 = 7 and d=−2d = -2, what is a20a_{20}? (A) −29-29 (B) −31-31 (C) −33-33 (D) −40-40

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Type 3

Build a sequence from two given terms

Divide (or subtract) the two given terms to isolate the ratio or difference, then back out a1a_1.

Example 3
A geometric sequence has a2=12a_2 = 12 and a5=96a_5 = 96. Find a1a_1 and rr. (A) a1=6,r=2a_1 = 6, r = 2 (B) a1=12,r=2a_1 = 12, r = 2 (C) a1=3,r=4a_1 = 3, r = 4 (D) a1=6,r=4a_1 = 6, r = 4

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Change in Arithmetic and Geometric Sequences | AP Precalculus — Mathfolis