Question

Certain number appear in both arithmetic progressions $17,21,25...$ and $16,21,26,...$ Find the sum of first twenty 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 ${T_m}^{\prime }=16+(m-1).5=5m+11.$

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 20 common terms

$=21+41+61+.....$ to 20 terms

$=\frac{20}{2}[2\times 21+(20-1).20]=4220.$