Σr=1n r = 1+2+...+n = n(n+1)/2. Sums of powers covered.

Sequences and series

Q: Recurrence vs closed-form?

un+1 = f(un) is recurrence. Closed-form gives un directly.

Arithmetic and geometric sequences, sums to n terms, sigma notation, recurrence.

Arithmetic: a, a+d, a+2d, ... Sum S_n = n(2a + (n-1)d)/2. Geometric: a, ar, ar², ... S_n = a(1-rⁿ)/(1-r). Sum to infinity exists for |r| < 1: S_∞ = a/(1-r).

Worked examples

Arithmetic 3, 7, 11, ...; n=10: S = 10(6 + 9×4)/2 = 210.

Geometric 2, 6, 18, ...; n=5: S = 2(1-243)/(1-3) = 242.

0.9 + 0.09 + 0.009 + ... = 0.9/(1-0.1) = 1.

Frequently asked questions

Σ (sigma) notation?

Σ_r=1ⁿ r = 1+2+...+n = n(n+1)/2. Sums of powers covered.

Recurrence vs closed-form?

u_n+1 = f(u_n) is recurrence. Closed-form gives u_n directly.