How many terms of the $A . P 17, 15, 13$ , ---- must be added to get the sum $72$ ? Explain the double answer.

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Solution

Sum $= ‘ 17$
Series is $17, 15, 13, . . . .$
$a = 17, d = - 2$
$s_{n} = \frac{n}{2} [2 (17) + (n - 1) - 2] 72 = \frac{n}{2} (34 - 2 n + 2) 144 = 36 n - 2 n^{2} n^{2} - 18 n + 72 = 0 n^{2} - 12 n - 6 n + 72 = 0 n (n - 12) - 6 (n - 12) = 0 n = 6, 12 a_{6} = 17 + 5 (- 2) = 7 a_{12} = 17 + 11 (- 2) = - 5$
Since it is decreasing AP
The sum gets increases til the positive terms are encountered and decreases since negitive values are added.

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