Question

An A.P. has even number of terms, the first term of which is unity. $S_1$ and $S_2$ stand for the sum of its first n terms and sum of its last $n$ terms such that $S_1/S_2=\lambda$ . What is the common difference of A.P.?

Answer 1

Let the common difference is $d$

And let number of terms is $2n$

$\therefore S_1=S_n=\frac{n}{2}[2+(n-1)d]$

and $S_2=S_{2n}=\frac{2n}{2}[2+(2n-1)d]$

Given $\frac{S_1}{S_2}=\frac{S_n}{S_{2n}-S_n}=\lambda$ $($ constant $)$

$\Rightarrow \frac{S_{2n}-S_n}{S_n}=\frac{1}{\lambda }=k$ $($ say $)$

$\Rightarrow \frac{S_{2n}}{S_n}-1=k\Rightarrow \frac{S_{2n}}{S_n}=k+1$

$\Rightarrow \frac{\frac{2n}{2}[2+(2n-1)d]}{\frac{n}{2}[2+(n-1)d]}=k+1$ $\Rightarrow \frac{2[2+(2n-1)d]}{[2+(n-1)d]}=k+1$

$\Rightarrow d=\frac{2(k-1)}{n(3-k)+k-1}$

Since ratio is independent of $n$ so coefficient of $n$ should be zero

$\Rightarrow 3-k=0\Rightarrow k=3$

So, $d=2$