If the sum of the coefficients of all even powers of x in the product
(1 + x + x2
+ ....+ x2n)(1 - x + x2 - x3 + ...... + x2n) is 61, then n is equal to _______.
Answer (integer)
30
Solution
(1 + x + x<sup>2</sup>
+ ....+ x<sup>2n</sup>)(1 - x + x<sup>2</sup> - x<sup>3</sup> + ...... + x<sup>2n</sup>)
<br><br>= a<sub>0</sub> + a<sub>1</sub>x + a<sub>2</sub>x<sup>2</sup>
+ …..
<br><br>put x = 1
<br><br> (2n + 1)$\times$1 = a<sub>0</sub> + a<sub>1</sub> + a<sub>2</sub> + …… (1)
<br><br>put x = –1
<br><br> 1$\times$(2n + 1) = a<sub>0</sub> – a<sub>1</sub> + a<sub>2</sub>+ …….. (2)
<br><br>Adding (1) and (2)
<br><br>4n + 2 = 2(a<sub>0</sub> + a<sub>2</sub> + ….. )
<br><br>$\Rightarrow$ 4n + 2 = 2 $\times$ 61
<br><br>$\Rightarrow$ n = 30
About this question
Subject: Mathematics · Chapter: Binomial Theorem · Topic: Binomial Expansion
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