Question

Certain numbers appear in both arithmetic progressions $17,21,25,$_______ and $16,21,26$,__________. Find the sum of first hundred numbers appearing in both progressions.

Answer 1

Denoting the nth and mth terms of the two progressions by $T_n$ and $T_m^{\prime }$, we have
$T_n=17+(n-1).4=4n+13$
and $T_m=16+(m-1).5=5m+11$.
For common terms, we must have
$T_n=T_m^{\prime }\Rightarrow 4n+13=5m+11$
$\Rightarrow 5m=2(2n+1)$
This shows that $2n+1=5k,k=1,3,5$,....
Hence the common terms are given by
$T_{2k}^{\prime }=5.2k+11=10k+11,k=1,3,5$,...
$\therefore$ Sum of first $100$ common terms
$=21+41+61+...$to $100$ terms
$=\frac{100}{2}[2\times 21+(100-1\left).20\right]=101100$.