JEE Main · 2015 · 04 Apr 2015 · Offline · Binomial Theorem

The sum of coefficients of integral powers of

x

in the binomial expansion of

(1 - 2 x)^{50}

is :

Answer: (D)

Solution

Expand

(1 - 2 x)^{50} = \sum_{r} (r 50) (- 2)^{r} x^{r /2}

; an integral power of

x

needs

r

even. A standard trick isolates those terms: setting

x = 1

gives the sum of all coefficients,

(1 - 2)^{50} = 1

, while evaluating

(1 + 2 x)^{50}

x = 1

gives

(1 + 2)^{50} = 3^{50}

, the same sum but with the signs of the odd terms flipped. Averaging the two keeps only the even-

r

(integral-power) terms:

\frac{1 + 3 ^{50}}{2} = \frac{1}{2} (3^{50} + 1)

JEE Main · 2015 · 04 Apr 2015 · Offline · Single correct option