Question

An arithmetic progression consists of $12$ terms whose sum is $354$.The ratio of the sum of the even terms to the sum of the odd terms is $32:27$. Find the common difference of the progression.

Answer 1

Sum of $12$ terms$=354$

$\Rightarrow \frac{12}{2}[2a+(12-1\left)d\right]=354\Rightarrow 6[2a+11d]=354\Rightarrow 2a+11d=59-\left(i\right)$

$\therefore \frac{Sumofeventerms}{Sumofoddterms}=\frac{32}{27}$

$\therefore$ Sum of even terms$=32x$ & Sum of odd terms$=27x$

$\therefore 32x+27x=354\Rightarrow 59x=354\Rightarrow x=6$

$\therefore$ Sum of even terms$=32\times 6=192$

$\Rightarrow t2+t4+t6+t8+t10+t12+192\therefore a_1=t2=a+d-\left(ii\right)$

Common difference$=t4-t2=a+3d-a-d=2d$

$\therefore$ Sum$=192$

$\Rightarrow \frac{n}{2}[2a+(n-1\left)d\right]=192\Rightarrow \frac{6}{2}\left[2\right(a+d)+5\times 2d]=192\Rightarrow [2a+2d+10d]=\frac{192}{3}=64\Rightarrow 2a+12d=64-\left(iii\right)$

Subtracting $\left(i\right)$ from $\left(iii\right)$

$d=5$

The common difference$=5$.