Question

Certain numbers appear in both arithmetic progressions $17,21$, . . . and $16,21$,........ Find the sum of the first hundred numbers appearing in both progressions.

Answer 1

$P1=17,21,25,29,33,37,41,....P2=16,21,26,31,36,41,......$

$\therefore$ Terms appearing in both $P1\&P2$ are $21,41,61,...$

where $a=21,d=20$

$\therefore S_{100}=\frac{100}{0}[2\times 21+(100-1\left)20\right]=50[42+1980]=50\times 2022=101100$