Question

If three successive coefficient in the expansion of ${(1+x)}^n$ are in A.P., then $(n+2)$ is

• A. at least $19$
• B. at most $19$
• C. a perfect square
• D. a perfect cube

Answer 1

Answer: (C)

a perfect square

Considering the three 3 consecutive terms ${}^n{C}_{r-1},{}^n{C}_r,{}^n{C}_{r+1}$ of $(1+x{)}^n$ to be in A.P, it must follow that
$(n-2r{)}^2=n+2$
Hence $n+2$ has to be a perfect square.