Computing Arithmetic Series

Pre-Calculus · Axiom Academy

EXAMPLE Computing Arithmetic Series Summing a finite arithmetic series with the average-of-the-ends formula. Find the sum of the arithmetic series . The first term is 5 , the terms go up by a fixed amount, and the last term is 62 . The 20 terms form a trapezoid. Its area — the average of the ends, (5 + 62)/2 , times the width 20 — is exactly the sum. Nice work — you summed a 20-term series without adding 20 numbers, by pairing each term with the average of the ends. Sum formula: — the number of terms times the average of the first and last term. Count the terms first: rearrange a_n = a_1 + (n-1)d to solve for n ; here gives n = 20 — don't forget the +1 . Sigma notation: the same idea sums — expand the first few terms to find a_1 , a_n , and n , then apply the formula. Real situations are arithmetic too: a theater with 15 seats in row 1 and 2 more each row sums the same way — count the rows, average the ends, multiply. Whenever a list of numbers climbs by a constant step, you can replace the long addition with one quick "average of the ends, times how many."

This is the written version of the interactive lesson above. See the full Pre-Calculus course.