Question

In the expansion of $(1+x{)}^n$ if the coefficients of three consecutive terms are in A.P., then $n+2$ is

• A. $n^2$
• B. a perfect square
• C. a perfect cube
• D. none

Answer 1

The correct option is B a perfect square

Let ${}^n{C}_{r-1},{}^n{C}_r,{}^n{C}_{r+1}$ be in AP
Therefore condition for the following terms to be in A.P is
${}^n{C}_{r-1}{+}^n{C}_{r+1}=2{\times }^n{C}_r$
On simplifying, we get,
$(n-2r{)}^2=n+2$ ...(i)
Hence, $n+2$ is perfect square.
Hence answer is option B.