worldofcalcsAP and GP Calculator
nth term (aₙ)
29
Sum of n terms (Sₙ)
155
Working
aₙ = a + (n − 1)d = 2 + (10 − 1) × 3 = 29
Sₙ = n/2 × [2a + (n − 1)d] = 10/2 × [2 × 2 + 9 × 3] = 155
First terms
2, 5, 8, 11, 14, 17, 20, 23, 26, 29
What are AP and GP?
An arithmetic progression (AP) adds the same number, the common difference d, to get each next term: 2, 5, 8, 11… A geometric progression (GP) multiplies by the same number, the common ratio r: 3, 6, 12, 24… This calculator finds the nth term and the sum of the first n terms of either, with the working as in Class 10 and 11 textbooks.
AP and GP formulas
AP: aₙ = a + (n − 1)d, Sₙ = n/2 × [2a + (n − 1)d] · GP: aₙ = arⁿ⁻¹, Sₙ = a(rⁿ − 1) ÷ (r − 1), S∞ = a ÷ (1 − r) when −1 < r < 1
a is the first term and n the number of terms. For an AP you can also use Sₙ = n/2 × (first term + last term).
Useful results
- The sum of the first n natural numbers is n(n + 1)/2, so 1 + 2 + … + 100 = 5,050.
- The sum of the first n odd numbers is n²: 1 + 3 + 5 + 7 = 16.
- Three numbers in AP can be written a − d, a, a + d; three in GP as a/r, a, ar. This simplifies many textbook problems.
- A GP with ratio between −1 and 1 shrinks towards 0, so its terms add up to a finite total: 1 + ½ + ¼ + … = 2.
- Compound interest is a GP: each year's amount is the previous one × (1 + rate).
Worked example: the AP 2, 5, 8, …
- First term a = 2 and common difference d = 3
- 10th term: a₁₀ = 2 + (10 − 1) × 3 = 29
- Sum of 10 terms: S₁₀ = 10/2 × [2 × 2 + 9 × 3] = 5 × 31 = 155
- Check with Sₙ = n/2 × (first + last) = 5 × (2 + 29) = 155
Sums of common series
Standard formulas and their values for n = 10 and n = 100.
| Series | Sum of n terms | n = 10 | n = 100 |
|---|---|---|---|
| Natural numbers 1, 2, 3, … | n(n + 1)/2 | 55 | 5,050 |
| Odd numbers 1, 3, 5, … | n² | 100 | 10,000 |
| Even numbers 2, 4, 6, … | n(n + 1) | 110 | 10,100 |
| Squares 1², 2², 3², … | n(n + 1)(2n + 1)/6 | 385 | 3,38,350 |
| Cubes 1³, 2³, 3³, … | [n(n + 1)/2]² | 3,025 | 2,55,02,500 |
The first three are APs; squares and cubes are not, but their sums are standard results.