Question

If $k$ is a positive integer such that $36+k,300+k,596+k$ are the squares of three consecutive terms of an arithmetic progression, then common difference of this $A.P.$ is

• A. $\pm 1$
• B. $\pm 2$
• C. $\pm 4$
• D. None of these

Answer 1

The correct option is C $\pm 4$

Let the 3 consecutive terms of AP be a - d, a, a + d

Given $(a-d{)}^2=36+k$ ______ (1)

$a^2=300+k$ __________ (2)

$(a+d{)}^2=596+k$ ________ (3)

(1) + (3) $\Rightarrow 2(a^2+d^2)=2k+632$

$a^2+d^2=k+316$ _____ (4)

from (2) & (4)

$300+k+d^2=k+316$

$\Rightarrow d^2=16\Rightarrow d=\pm 4$