Special Sum Formulas

Algebra 2 · Axiom Academy

Closed forms for , , and — and the pictures that prove them. 1. The Constant Sum: adding the same number The gentlest sum adds one value over and over. Add the constant c a total of n times and you are just counting n equal groups of c — so the answer is the area of an n -by- c rectangle . Setting c = 1 recovers the plain count . Repeated addition of a constant The special case c = 1 : just counting Legend says a young Carl Friedrich Gauss added in seconds. His trick: write the sum forwards and backwards and add — every matched pair totals n+1 . Two copies of the staircase — one rising , one falling — interlock into a solid rectangle. So twice the sum is n(n+1) , and the sum itself is half of that. Two staircases fill the rectangle So the sum is half the rectangle Evaluate without adding term by term Squares grow faster. Stack a square, then a , on up to an square, and add their areas. The running totals are captured by one tidy formula with three factors. and directly, 1 + 4 + 9 + 16 + 25 = 55 — the formula and the count agree. The most striking of all: the sum of cubes equals the square of the sum of the first n numbers. A single square whose side is splits into L-shaped shells whose areas are exactly . Why the shells work: the k -th L-shell wraps a square of side out to side S_k = S_ k-1 +k , so its area is . and directly, 1 + 8 + 27 + 64 = 100 = 10^2 — the square of 1+2+3+4 .

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