First slide

Arithmetic progression

Question

The maximum sum of the series $20 + 19 \frac{1}{3} + 18 \frac{2}{3} + 18 + \dots$ is

Moderate

A

310
B

290
C

320
D

None of these

Solution

The given series is arithmetic whose first term = 20, common difference = – . As the common difference is negative, the terms will become negative after some stage. So the sum is maximum if only positive terms are added.
Now, $\begin{array}{l} t_{n} = 20 + (n - 1) (- \frac{2}{3}) \geq 0 if 60 - 2 (n - 1) \geq 0 \\ or 62 \geq 2 n or 31 \geq n \end{array}$
$∴$ The first 31 terms are non-negative.
$\begin{array}{l} ∴ maximum sum = S_{31} = \frac{31}{2} [2 \times 20 + (31 - 1) (- \frac{2}{3})] \\ = \frac{31}{2} (40 - 20) - 310 \end{array}$

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