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The sum of the coefficients of all even degree terms is x in the expansion of (x + sqrt(x^(3) - 1))^(6) + (x - sqrt(x^(2) - 1))^(6), (x gt 1) is equal to |
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Answer» 29 `(a + b)^(N) + (a - b)^(n) =` `2[.^(n)C_(0) a^(n) + .^(n)C_(2) a^(n - 2) b^(2) + .^(n)C_(4) a^(n - 1) b^(4)` + …… 1 Given expansion is `(x + SQRT(x^(3) - 1))^(6) + (x - sqrt(x^(3) - 1))^(6)` `= 2[.^(6)C_(0) x^(6) + .^(6)C_(2) x^(4) (sqrt(x^(2) - 1))^(2)` `{ :' (a + b)^(n) + (a - b)^(n)` `= 2 [.^(n)C_(0) a^(n) + .^(n)C_(2) a^(n - 2) b^(2) + .^(n)C_(4)a^(n - 4) b^(4) + ...]}` The SUM of the terms with EVEN power of x. ` 2[.^C_0x^6+.^C_2(-x^4)+.^6C_4x^8+.^6C_4x^2+.^6C_6(-1-3x^6)]`. ` =2[.^6C_0x^6-.^6C_2x^4+.^6C_4x^8 +.^6C_4x^2-1 -3x^6]` Now, the required sum of the coefficient of even powers of x in `(x+sqrt(x^3-1))^6+(x-sqrt(x^3-1))^6` ` =2[.^6C_0-.^6C_2+.^6C_4+.^6C_4-1-3]` ` =2[1-15+15+15-1-13]=2(15-3]=2(15-3)=24` |
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