(A) : If the three successive coefficients in ${(1 + x)}^{n}$ are 6, 15, 20 then n = 8.
(R) : If coefficient of $r^{th}$ , ${(r + 1)}^{th}$ , ${(r + 2)}^{th}$ terms in ${(1 + x)}^{n}$ are in A.P. then $n + 2 = {(n - 2 r)}^{2} .$

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Both A and R are true and R is correct
explanation of A

Both A and R are true but R is not correct
explanation of A

A is true, R is false

A is false, R is true

answer is D.

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Detailed Solution

(A)

$\begin{array}{rcl} {}^{n}{C_{r - 1}} & = & 6_{1} {}^{n}{C_{r}} = 15_{1} {}^{n}{C_{r + 1}} = 20 \\ \frac{{}^{n}{C_{r}}}{{}^{n}{C_{r - 1}}} & = & \frac{15}{6} \Rightarrow \frac{n - r + 1}{r} = \frac{5}{2} \\ 2 n - 2 r + 2 & = & 5 r \\ 2 n + 2 = 7 r \\ \frac{{}^{n}{C_{r + 1}}}{{}^{n}{C_{r}}} & = & \frac{20}{15} \Rightarrow \frac{n - r - 1 + 1}{r + 1} = \frac{4}{3} \\ 3 n - 3 r & = & 4 r + 4 \\ 3 n - 4 = 7 r \end{array}$