Find the first 3 terms and the sum to infinity of a geometric series with first term, 10 and common ratio 0.2

General form of a Geometric series: a + ar + ar^2 + ar^3 + ar^4 + … First three terms: 10 + 100.2 + 100.2^2 = 10 + 0.2 + 0.04Nth Term of a Geometric Series: ar^n-1 where there are n terms Proof of sum of terms: S = a + ar + ar^2 + ar^3 + … + ar^(n-1) Multiply by r: Sr = ar + ar^2 + ar^3 + … + ar^n – (2) : S – Sr = (a + ar + ar^2 + ar^3 + … + ar^(n-1)) (ar + ar^2 + ar^3 + … + ar^(n1) + ar^n) Terms: ar, ar^2, … ar^(n-1) cancel out so: S – Sr = a – ar^n Grouping the terms S and a: S(1-r) = a(1-r^n)              ->           S = a(1-r^n)/(1-r) Which is the sum of a geometric series. To find the sum to infinity, we set n = infinity. As n becomes larger and larger, closer to infinity (tending). R^n becomes smaller and smaller since r < 1. Where n tends to infinity, r^n tends to 0 so: S = a(1-0)/(1-r)                ->           S = a/(1-r) Where a = 10 and r = 0.2: S = 10/(1-0.2) = 10/0.8 = 12.5