Question

The total number of terms in the expansion of (x + a)⁵¹ – (x – a)⁵¹ after simplification is
(a) 102
(b) 25
(c) 26
(d) none of these

Answer 1

$$\begin{gathered} \mathrm{for}{(x+a)}^{51}–{(x–a)}^{51} \\ \mathrm{Since}{(x+a)}^{51}={}^{51}{C}_0{x}^{51}{+}^{51}{C}_1{x}^{50}a{+}^{51}{C}_2{x}^{49}{a}^2+....{+}^{51}{C}_{51}{a}^{51} \\ \mathrm{and}{(x-a)}^{51}={}^{51}{C}_0{x}^{51}{-}^{51}{C}_1{x}^{50}a{+}^{51}{C}_2{x}^{49}{a}^2....{}^{51}{C}_{51}{a}^{51} \\ \mathrm{Subtracting}\mathrm{above}\mathrm{values}, \\ {(x+a)}^{51}-{(x-a)}^{51}=2\left({}^{51}C_1{x}^{50}a+{}^{51}C_{3}x{}^{48}a^3+...+{}^{51}C_{51}{a}^{51}\right) \end{gathered}$$
i.e. 1^st, 3^rd, 5^th, 7^th ________ 49^th, 51^th term are there
∴ applying A.P.
a + ( n – 1)d = 51
i.e. 1 + (n – 1) 2 = 51
i.e. 2(n – 1) = 50
i.e. n = 26

Hence, the correct answer is option C.