Predicting What Comes Next : Exploring Sequences and Progressions — Formula Sheet

u_n = 2n - 1

s_n = 5n - 2

t_n = a + (n - 1) × d

t_n = ar^{n-1}

S_n = n(n + 1)/2

t_1 = a, t_n = t_{n-1} + d for n ≥ 2

V_1 = 1, V_2 = 2, V_n = V_{n-1} + V_{n-2} for n ≥ 3

t_1 = 1, t_n = 3 × t_{n-1} for n ≥ 2

s_1 = 1, s_n = (3/4) × s_{n-1} for n ≥ 2

General form of an AP: a, a + d, a + 2d, a + 3d, ... , a + (n - 1) × d

For u_n = 2n - 1, the 137th term is found by solving 2n - 1 = 137, giving n = 69.

For an AP, the recursive rule is t1 = a, tn = tn-1 + d for n ≥ 2.