AP vs GP: Sequences and Series for CA Foundation Maths
Sequences power many financial calculations — EMIs use AP concepts, compound growth uses GP. Mastering both unlocks marks across Business Maths.
head-to-Head Comparison
| Basis | Arithmetic Progression (AP) | Geometric Progression (GP) |
|---|---|---|
| Key Constant | Common Difference (d) — added to each term | Common Ratio (r) — multiplied into each term |
| General Term (nth term) | Tₙ = a + (n − 1)d | Tₙ = a × r^(n−1) |
| Sum of n Terms | Sₙ = n/2 × [2a + (n−1)d] or n/2 × (a + l) | Sₙ = a(rⁿ − 1)/(r − 1) for r ≠ 1; Sₙ = na for r = 1 |
| Growth Pattern | Linear growth (equal increments) | Exponential growth (multiplicative increments) |
| Real-World Link | Salary increments of fixed amount; arithmetic mean | Compound interest growth; population growth; depreciation (WDV) |
The 'Sum to Infinity' Trap
The sum to infinity exists only for a GP where |r| < 1. Formula: S∞ = a/(1 − r). An AP has NO sum to infinity (it diverges). A common exam trick is to give a GP with |r| < 1 and ask for S∞ — don't apply the finite sum formula.
Common Ground (Similarities)
- Both are sequences with a definite pattern that can be described by a formula.
- Both require knowledge of the first term (a) and the common element (d or r) to find any term.
- Both have sum formulas used extensively in financial mathematics and actuarial science.
Test Your Understanding
Q1: The 7th term of the AP 3, 7, 11, 15... is:
27 ✅
23
25
29
Explanation: a = 3, d = 4. T₇ = 3 + (7−1) × 4 = 3 + 24 = 27.
Q2: The sum to infinity of GP 1, 1/2, 1/4, 1/8,... is:
1
2 ✅
4
∞
Explanation: a = 1, r = 1/2 (|r| < 1). S∞ = 1/(1 − 1/2) = 1/(1/2) = 2.