SATHEE: Binomial Theorem 1 Question 16

Binomial Theorem 1 Question 16

17. If the number of terms in the expansion of $1 - \frac{2}{x} + {\frac{4}{x^{2}}}^{n}, x \neq 0$ , is 28 , then the sum of the coefficients of all the terms in this expansion, is

(a) 64

(b) 2187

(d) 729

(2016 Main)

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Answer:

Correct Answer: 17. (5)

Solution:

Clearly, number of terms in the expansion of

$\begin{array}{lc} 1 - \frac{2}{x} + {\frac{4}{x^{2}}}^{n} is \frac{(n + 2) (n + 1)}{2} or^{n + 2} C_{2} . \\ [assuming \frac{1}{x} and \frac{1}{x^{2}} distinct] \\ ∴ & \frac{(n + 2) (n + 1)}{2} = 28 \\ \Rightarrow (n + 2) (n + 1) = 56 = (6 + 1) (6 + 2) \Rightarrow n = 6 \end{array}$

Hence, sum of coefficients $= (1 - 2 + 4)^{6} = 3^{6} = 729$ Note As $\frac{1}{x}$ and $\frac{1}{x^{2}}$ are functions of same variables, therefore number of dissimilar terms will be $2 n + 1$ , i.e. odd, which is not possible. Hence, it contains error.

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Binomial Theorem 1 Question 16

17. If the number of terms in the expansion of 1−2x+4x2n,x≠0, is 28 , then the sum of the coefficients of all the terms in this expansion, is

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17. If the number of terms in the expansion of $1 - \frac{2}{x} + {\frac{4}{x^{2}}}^{n}, x \neq 0$ , is 28 , then the sum of the coefficients of all the terms in this expansion, is