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310 ALGEBRA. 379. The Sum of n Terms in G. P. Let a be the first term, r the common ratio, n the number of terms, and S the sum required. Then S = a + ar + ar2 + ... + ar n-2 + ar n-1 ; multiplying every term by r, we have

rS = ar + ar2 + ... + ar n-1 + ar n. Hence by subtraction, rS — S = ar n — a ; (r- 1)S = a(r n - 1); S = a(r n - 1) r-1 (1)

Changing the signs in numerator and denominator S = a(1 -r n)1 -r (2).

Note. It will be found convenient to remember both forms given above for S, using (2) in all cases except when r is positive and greater than 1.

Since ar n - 1 = l, it follows that ar n = rl, and formula (1) may be written

S = rl - a r — 1

Ex. 1. Sum the series 81, 64, 86, ... to 9 terms.

The common ratio = |i = |, which is less than 1;

hence the sum = ^^l - (|) !} ^ 243{1 - (f )9}

Ex. 2. Sum the series 2 3, -1, 3 2, ... to 7 terms.

The common ratio = — 3 2 ; hence by formula (2) 1 + f I.

EXAMPLES XXXIV. c.

1. Find the 5th and 8th terms of the series 3, 6, 12, .... 2. Find the 10th and 16th terms of the series 256, 128, 64, .... 3. Find the 7th and 11th terms of the series 64, -32, 16, ....