The sum of all those terms, of the arithmetic progression 3, 8, 13,........373, which are not divisible by 3 , is equal to: Sequence & Series (Mathematics)

The given A.P. is $3,8,13, \ldots, 373$ $\\T_n=a+(n-1) d \\$ $\Rightarrow 373=3+(n-1) 5 \Rightarrow n=75 \\$ $S_n=\frac{75}{2}[3+373]=14100\\$ Numbers which are divisible by 3 are $3,18, \ldots, 363.$ $\\ T_n^{\prime}=3+\left(n^{\prime}-1\right) 15 \\$ $\Rightarrow 363=3+15 n^{\prime}-15 \Rightarrow 15 n^{\prime}=375$ $\\ \Rightarrow n^{\prime}=25 \\$ $S_n^{\prime}=\frac{25}{2}[3+363]=4575 \\$ $\therefore$ { Required sum }= $14100-4575=9525$

Explanation

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