InterviewSolution
Saved Bookmarks
InterviewSolution
| 1. |
Thesum of coefficients of integral powers of x in the binomial expansion of `(1-2sqrt(x))^(50)`is:(1) `1/2(3^(50)+1)`(2) `1/2(3^(50))`(3) `1/2(3^(50)-1)`(4) `1/2(2^(50)+1)`A. `(1)/(2) (3^(50) + 1)`B. `(1)/(2) (3^(50))`C. `(1)/(2) (3^(50) - 1)`D. `(1)/(2) (3^(50) + 1)` |
|
Answer» Let `T_(t + 1)` be the general term in the expansion of `(1 - 2 sqrt(x))^(50)` `:. T_(r + 1) = .^(50)C_(r ) (1)^(50 - r) (-2x^(1//2))^(r ) = .^(50)C_(r ) 2^(r ) x^(r//2) (-1)^(r )` For the integral power of x,r should be even integer. `:.` Sum of coefficientss `= underset(r = 0)overset(25)(sum) .^(50)C_(2r) (2)^(2r)` `= (1)/(2) [(1 + 2)^(50) + (1 - 2)^(50)] = (1)/(2) (30^(50) + 1)` Alternate solution we have `(1 - 2 sqrt(2))^(50) = C_(0) - C_(1) 2 sqrt(x) + C_(2) (sqrt(2x))^(2 + ... + C_(50) (2 sqrt(x))^(50)` .... (i) `(1 + 2 sqrt(x))^(50) = C_(0) + C_(1) 2 sqrt(x) + C_(2) (2 sqrt(x))^(2) + ... + C_(50) (2 sqrt(x))^(50)` ..... (iii) On adding Eqs. (i) and (ii) we get `(1 - 2sqrt(x))^(50) + (1 + 2 sqrt(x))^(50)` `= 2 [C_(0) + C_(2) (2 sqrt(x))^(2) + ... + C_(50) (2 sqrt(x))^(50)]` `implies ((1 - 2 sqrt(x))^(50) + (1 + 2 sqrt(1))^(50))/(2)` `= C_(0) + C_(2) (2 sqrt(x))^(2) + .... + C_(50) (2 sqrt(x))^(50)` On putting x - 1, we get `((1 - 2 sqrt(1))^(50) + (1 + 2 sqrt(1))^(50))/(2) = C_(0) + C_(2) + ... + C_(50) (2)^(50)` `implies ((-1)^(50) + (3)^(50))/(2) = C_(0) + C_(2) (2)^(2) + ... + C_(50) (2)^(50)` `implies (1 + 3^(50))/(2) = C_(0) + C_(2) (2)^(2) +... + C_(50) (2)^(50)` . |
|