What are AP and GP?
Arithmetic Progression (AP) is a sequence where the difference between consecutive terms is constant (common difference r). Geometric Progression (GP) is a sequence where the quotient between consecutive terms is constant (common ratio q).
AP: aₙ = a₁ + (n − 1) × r
Sₙ = n × (a₁ + aₙ) ÷ 2
GP: aₙ = a₁ × q^(n−1)
Sₙ = a₁ × (qⁿ − 1) ÷ (q − 1) [q ≠ 1]
Sₙ = n × a₁ [q = 1]Types of AP
- Increasing: r > 0 → terms grow (2, 5, 8, 11…)
- Decreasing: r < 0 → terms shrink (10, 7, 4, 1…)
- Constant: r = 0 → all terms equal (5, 5, 5…)
Types of GP
- Increasing: q > 1 and a₁ > 0 (1, 2, 4, 8…)
- Decreasing: 0 < q < 1 and a₁ > 0 (100, 50, 25…)
- Alternating: q < 0 → terms alternate in sign (1, −2, 4, −8…)
Frequently asked questions
How do I tell whether a sequence is an AP or a GP?
For an AP, subtract consecutive terms — the difference must be the same (r). For a GP, divide consecutive terms — the quotient must be the same (q). If neither, it is not a simple AP or GP.
What is the story of Gauss and the AP?
The mathematician Gauss, as a child, computed 1+2+...+100=5050 by noticing that symmetric pairs can be added: (1+100)+(2+99)+...=50×101=5050. That is the basis of the AP sum formula.
What is a GP used for in practice?
A GP models exponential growth: compound interest (the balance doubles each period), population growth, radioactive decay and any phenomenon that multiplies by a constant rate.
What is a convergent infinite GP?
When |q| < 1, the sum of infinitely many terms of the GP converges: S∞ = a₁ ÷ (1 − q). It is used in mathematical series and in modeling physical processes.