Question:
If the number of terms in the expansion of $\left( 1 - \frac{2}{x} + \frac{4}{x^2} \right)^n , x \neq 0$ , is $28$ , then the sum of the coefficients of all the terms in this expansion, is :
If the number of terms in the expansion of $\left( 1 - \frac{2}{x} + \frac{4}{x^2} \right)^n , x \neq 0$ , is $28$ , then the sum of the coefficients of all the terms in this expansion, is :
Updated On: Aug 14, 2024
- 64
- 2187
- 243
- 729
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The Correct Option is D
Solution and Explanation
Number of terms $ = \frac{(n +1)(n+2)}{2} = 28 $
$\Rightarrow n = 6$
$\therefore \, \, a_0 + \frac{a_1}{x} + \frac{a_2}{x^2} + .... + \frac{a_2n}{x^{2n}} = \left( 1 - \frac{2}{x} + \frac{4}{x^2} \right)^n$
Put $x = 1, n = 6 , a_0 + a_1 + a_2 + ... + a_{2n} = 3^6 = 729$
$\Rightarrow n = 6$
$\therefore \, \, a_0 + \frac{a_1}{x} + \frac{a_2}{x^2} + .... + \frac{a_2n}{x^{2n}} = \left( 1 - \frac{2}{x} + \frac{4}{x^2} \right)^n$
Put $x = 1, n = 6 , a_0 + a_1 + a_2 + ... + a_{2n} = 3^6 = 729$
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Concepts Used:
Binomial Theorem
The binomial theorem formula is used in the expansion of any power of a binomial in the form of a series. The binomial theorem formula is
Properties of Binomial Theorem
- The number of coefficients in the binomial expansion of (x + y)n is equal to (n + 1).
- There are (n+1) terms in the expansion of (x+y)n.
- The first and the last terms are xn and yn respectively.
- From the beginning of the expansion, the powers of x, decrease from n up to 0, and the powers of a, increase from 0 up to n.
- The binomial coefficients in the expansion are arranged in an array, which is called Pascal's triangle. This pattern developed is summed up by the binomial theorem formula.