Question

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?

• A. Absolute difference of all possible values of $k$ is $\frac{2}{3}$.
• B. The sum of all possible values of $k$ is $\frac{4}{3}$.
• C. If the number of terms of arithmetic progression is 20 , then the sum of all terms of all possible arithmetic progressions is 420 .
• D. The number of possible non-zero values of common difference of arithmetic progressions is 1 .

Answer 1

Answer: (A), (B), (C), (D)

Let number of terms $=2n$
$\text{ So, }k=\frac{S_n}{S_{2n}-S_n}=\frac{\frac{n}{2}[2a+(n-1)d]}{\frac{2n}{2}[2a+(2n-1)d]-\frac{n}{2}[2a+(n-1)d]}=\frac{na+\frac{n(n-1)d}{2}}{na+\frac{n(3n-1)d}{2}}(a=1)$
$k=\frac{2+(n-1)d}{2+(3n-1)d}=\frac{(2-d)+nd}{(2-d)+3nd}$
Hence, either $d=2$ or $d=0$
if $d=0$, then $k=1$ and if $d=2$, then $k=\frac{1}{3}$.