Pattern Recognition
Half of CAT sequence questions are won the moment you name the pattern. The first move is always to write out three or four terms and look at the differences. A constant first difference means an AP (linear in n); a constant second difference means a quadratic in n (so the term is an² + bn + c, and you can fit a, b, c from three terms). A constant ratio means a GP. If neither the differences nor the ratios settle, check whether the term is built from a known special sum — partial sums of 1,2,3,... give the triangular numbers 1,3,6,10,...; partial sums of odd numbers give the perfect squares 1,4,9,16,...; partial sums of cubes give perfect squares of triangular numbers. The CAT-smart habit is to always ask "is this the nth term, or a sum up to n?" because the same surface pattern hides very different closed forms. Once you have a closed form, plug in n to verify against the terms you wrote — a thirty-second check that saves a wrong answer.
✅ Solved examples
✏️ Practice — try these, take hints as needed
📝 Topic test — 8 questions
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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 |