The sum of all those terms, of the arithmetic progression 3, 8, 13,..., 373, which are not divisible by 3, is equal to __________ . [2023]

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Solution

Q.
The sum of all those terms, of the arithmetic progression 3, 8, 13,..., 373, which are not divisible by 3, is equal to __________ . [2023]

Ans.
(9525)

$The given A.P. is 3, 8, 13, \dots, 373$

$T_{n} = a + (n - 1) d$

$373 = 3 + (n - 1) 5$ $\Rightarrow n = 75$

$S_{n} = \frac{75}{2} [3 + 373] = 14100$

$Numbers which are divisible by 3 are 3, 18, \dots, 363 .$

$T'_{n} = 3 + (n' - 1) 15$

$\Rightarrow 363 = 3 + 15 n' - 15 \Rightarrow 15 n' = 375 \Rightarrow n' = 25$

$S'_{n} = \frac{25}{2} [3 + 363] = 4575$

$∴$ $Required sum = 14100 - 4575 = 9525$

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