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Sequences and Series

Sequences and Series​

Arithmetic progression:

nth term: a_n = a + (n-1)d
Sum: S_n = n/2 * [2a + (n-1)d]
Also: S_n = n/2 * (first + last)

Geometric progression:

nth term: a_n = ar^(n-1)
Sum: S_n = a(r^n - 1)/(r - 1), r != 1
Infinite sum: S_inf = a/(1-r), if |r| < 1

Common sums:

1 + 2 + ... + n = n(n+1)/2
1^2 + 2^2 + ... + n^2 = n(n+1)(2n+1)/6
1^3 + 2^3 + ... + n^3 = [n(n+1)/2]^2

Speed tricks:

  • Check if differences are constant: AP.
  • Check if ratios are constant: GP.
  • For repeating decimals, use GP.