Question:
If $(2x^2 - x - 1)^5 = a_0 + a_1x + a_2x^2 + ... + a_{10}x^{10}$, then,
$a_2 + a_4 + a_6 + a_8 + a_{10} =$
If $(2x^2 - x - 1)^5 = a_0 + a_1x + a_2x^2 + ... + a_{10}x^{10}$, then,
$a_2 + a_4 + a_6 + a_8 + a_{10} =$
Updated On: Jul 6, 2022
- $15$
- $30$
- $16$
- $17$
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The Correct Option is D
Solution and Explanation
$(2x^2 - x - 1)^5 = a_0 + a_1x + a_2x^2 + ... + a_{10}x^{10}$
Put $x = 1$, we get, $0 = a_0 + a_1 + a_2 + ... + a_{10} \quad ...(i)$
Put $x = - 1$, we get, $32 = a_0 - a_1 + a_2 - ... + a_{10} \quad ...(ii)$
Adding $(i)$ and $(ii)$, we get
$a_2 + a_4 + ... + a_{10} = (32/2) + 1$
$ = 16 + 1 = 17 \,\,(\because a_0 = - 1)$
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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.