Q. An arithmetic progression has the following property: For an even number of terms, the ratio of the sum of first half of the terms to the sum of second half is always equal to a constant ' $k$ '. Let the first term of arithmetic progression is 1 . Then which of the following statement(s) is(are) correct?
Sequences and Series
Solution:
Let number of terms $= 2n $
$\text { So, } k=\frac{S_n}{S_{2 n}-S_n}=\frac{\frac{n}{2}[2 a+(n-1) d]}{\frac{2 n}{2}[2 a+(2 n-1) d]-\frac{n}{2}[2 a+(n-1) d]}=\frac{n a+\frac{n(n-1) d}{2}}{n a+\frac{n(3 n-1) d}{2}} \quad(a=1) $
$k=\frac{2+(n-1) d}{2+(3 n-1) d}=\frac{(2-d)+n d}{(2-d)+3 n d}$
Hence, either $d=2$ or $d=0$
if $d =0$, then $k =1$ and if $d =2$, then $k =\frac{1}{3}$.
