1.

If `(1 + x + x^(2) + x^(3))^(n)= a_(0) + a_(1)x + a_(2)x^(2) + a_(3) x^(3) +...+ a_(3n) x^(3n)`, then the value of `a_(0) + a_(4) +a_(8) + a_(12)+….. ` isA. -1B. 0C. `4^(n-1)`D. n

Answer» Correct Answer - c
We have,
` a_(0) + a_(1)x + a_(2)x^(2) + a_(3) x^(3) +...+ a_(3n) x^(3n)=(1 + x + x^(2) + x^(3))^(n)`
`rArr a_(0) + a_(2)x^(2) + a_(4)x^(4) +...)+( a_(1) x + a_(3) x^(3)+...) =(1 + x + x^(2) + x^(3))^(n)`
Puttinhg x = 1, - 1, i and -i respectively, we get
`( a_(0) + a_(2) + a_(4) +...)+( a_(1) + a_(3) +a_(5)+...) =4^(n)` ....(i)
`( a_(0) + a_(2) + a_(4) +...) - ( a_(1) + a_(3) +a_(5)+...) = 0 ` ...(ii)
`( a_(0) - a_(2) + a_(4) - a_(6) +...)+i ( a_(1) - a_(3) +a_(5)+...) = 0 ` ...(iii)
`( a_(0) - a_(2) + a_(4) - a_(6) +...)- i ( a_(1) - a_(3) +a_(5)+...) = 0 ` ....(iv)
Adding (i) and (ii), we get
` 2 (a_(0) + a_(2) + a_(4) + ....) = 4 ^(n)` ....(v)
Adding (iii) and (iv), we get
`2 (a _(0) - a_(2) + a_(4) - a_(6)+ ....) = 0` ...(vi)
Adding (v) and (vi), we get
` 4 ( a_(0) + a_(4) + a_(8) + ....) = 4^(n) rArr a_(0) + a_(4) + a_(8) + ... = 4^(n-1)`


Discussion

No Comment Found

Related InterviewSolutions