Question

The sum of first six terms of an arithmetic progression is 42. The ratio of its 10th term to its ${30}^{th}$ term is 1 : 3 calculate the first and ${13}^{th}$ term of the A.P.

Answer 1

$S_6=42$ $\frac{7_{10}}{7_{30}}=\frac{1}{3}$

$42=\frac{6}{2}[2a+5d]\Rightarrow 2a+5d=14$

$\frac{a+9d}{a+29d}=\frac{1}{3}\Rightarrow 3a+27d=a+29d$

$\Rightarrow 2a=2d$

$\Rightarrow a=d$

2a+5d =14

$\Rightarrow 2d+5d=14\Rightarrow d=a=2$

$7_1=a=2$

$7_{13}=a+12d=2+24=26$