Arithmetic
Constant difference: add or subtract d.
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GMAT Quant
Geometric and Alternating Sequences
Recognize constant-ratio sequences, compute geometric sums, and handle simple alternating patterns.
Definition
A geometric sequence multiplies by a constant ratio r.
Unlike an arithmetic sequence, the change is multiplicative rather than additive.
Nth term
The nth term is a_n = a1*r^(n-1).
You multiply by r exactly n - 1 times to move from the first term to the nth.
Example: 3, 6, 12, 24, ...
- First term
- 3
- Ratio
- 2
- 5th term
- 3*2^4
- Result
- 48
Geometric sum
The sum of the first n terms uses the geometric-sum formula.
For r != 1: S_n = a1*(r^n - 1)/(r - 1).
Example: 2 + 6 + 18 + 54
- a1
- 2
- r
- 3
- n
- 4
- Sum
- 2*(3^4 - 1)/(3 - 1) = 80
Arithmetic and geometric sequences signal different operations.
Geometric
Constant ratio: multiply or divide by r.
Alternating pattern
Alternating sequences often collapse in pairs.
For 1, -1, 1, -1, ... each pair sums to zero. An odd number of terms leaves one extra 1.
Mini application
Try this: sum the first 12 terms of 1, -1, 1, -1, ...
Pair the terms before scrolling.
Solution
The sum is 0.
Twelve terms make six pairs, and each pair sums to zero.
Remember this
First ask: constant difference, constant ratio, or repeating pattern?
Classification determines the efficient sequence tool.