Special Series
Three identities cover almost every CAT summation. Sum of the first n naturals is n(n+1)/2; sum of the first n squares is n(n+1)(2n+1)/6; sum of the first n cubes is [n(n+1)/2]², which is exactly the square of the first identity — a fact CAT loves to hide. The professional move when you see a sum like Σ(k² + 3k) is to split it by linearity into Σk² + 3Σk and apply the identities separately, rather than adding term by term. Telescoping is the second pillar: when a term is a fraction like 1/[k(k+1)], rewrite it as 1/k − 1/(k+1) so that consecutive pieces cancel and only the first and last survive — turning a long sum into a two-term subtraction. The general split 1/[k(k+d)] = (1/d)[1/k − 1/(k+d)] handles spacing-d denominators. Spotting telescoping early is the single biggest time-saver in this chapter; whenever a sum has products in the denominator, suspect it.
✅ Solved examples
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Formula Reference Sheet
Special sums (first n terms)
| Sum of first n naturals | 1+2+...+n = n(n+1)/2 |
|---|---|
| Sum of first n squares | 1²+2²+...+n² = n(n+1)(2n+1)/6 |
| Sum of first n cubes | 1³+2³+...+n³ = [n(n+1)/2]² |
| Sum of first n odd numbers | 1+3+5+...+(2n−1) = n² |
| Sum of first n even numbers | 2+4+...+2n = n(n+1) |
| Key link: cubes = (sum)² | Σk³ = (Σk)² = [n(n+1)/2]² |
Telescoping, AGP & recurrences
| Telescoping split | 1/[k(k+1)] = 1/k − 1/(k+1) |
|---|---|
| General telescoping | 1/[k(k+d)] = (1/d)[1/k − 1/(k+d)] |
| AGP sum to infinity (|r|<1) | S∞ = a/(1−r) + dr/(1−r)² |
| AGP finite sum method | Compute S − rS to collapse to a GP |
| Linear recurrence (Fibonacci) | F(n) = F(n−1) + F(n−2) |
| Sum of first n Fibonacci | F1+F2+...+Fn = F(n+2) − 1 |