If $(1 + x + 2 x^{2})^{20} = a_{0} + a_{1} x + a_{2} x^{2} + . . . . a_{40} x^{40}$ . Find the values of $a_{1} + a_{3} + a_{5} + . . . . a_{39}$ .

Open in App

Solution

We have,

${(1 + x + 2 x^{2})}^{20} = a_{0} + a_{1} x + a_{2} x^{2} + a_{3} x^{3} + . . . . . . . . . + a_{40} x^{40}$

Let, put q $x = 1$ and we get,

${(1 + 1 + 2 {(1)}^{2})}^{20} = a_{0} + a_{1} + a_{2} + a_{3} + . . . . . . . . + a_{40}$

$4^{20} = a_{0} + a_{1} + a_{2} + a_{3} + . . . . . . . . + a_{40} . . . . . . (1)$

Again, put

$x = - 1$ and we get,

${(1 - 1 + 2 {(1)}^{2})}^{20} = a_{0} - a_{1} + a_{2} - a_{3} + . . . . . . . . + a_{40}$

$2^{20} = a_{0} - a_{1} + a_{2} - a_{3} + . . . . . . . . + a_{40} . . . . . . (2)$

On subtracting equation (1) and (2) to, we get,

$4^{20} - 2^{20} = 2 (a_{1} + a_{3} + a_{5} + . . . . . . . + a_{39})$

$\Rightarrow (a_{1} + a_{3} + a_{5} + . . . . . . . + a_{39}) = \frac{4^{20} - 2^{20}}{2}$

$\Rightarrow (a_{1} + a_{3} + a_{5} + . . . . . . . + a_{39}) = \frac{2^{20} (2^{20} - 1)}{2}$

$\Rightarrow (a_{1} + a_{3} + a_{5} + . . . . . . . + a_{39}) = 2^{19} (2^{20} - 1)$
Hence, this is the answer.

Suggest Corrections

Similar questions

Q. If

{(1 + x - 2 x^{2})}^{20} = a_{0} + a_{1} x + a_{2} x^{2} + \dots + a_{40} x^{40}

and the value of

a_{1} + a_{3} + a_{5} + \dots + a_{39} = - 2^{k}

, then

k =

If $(1 + x + 2 x^{2})^{20} = a_{0} + a_{1} + a_{2} x^{2} + . . . + a_{40} x^{40}, t h e n a_{1} + a_{3} + a_{5} + . . . + a_{37} e q u a l s$ :

$(1 + x + 2 x^{2})^{20}$ = $a_{0} + a_{1} x + a_{2} x^{2}$ ................... $a_{40} x^{40}$ , If $a_{0} + a_{1} + a_{2} + a_{3} . . . . . . . . . . . . . . . . . . a_{40} = 2^{a}$

$a_{0} - a_{1} + a_{2} - a_{3} . . . . . . . . . . . . . . . . . . . . . . a_{40} = 2^{b}$ . Find the value of a + b

___

Q. If

{(1 + x + 2 x^{2})}^{20} = a_{0} + a_{1} x + a_{2} x^{2} + . . . . . + a_{10} x^{40}

, then

a_{1} + a_{3} + a_{5} + . . . . . . + a_{39}

equals

If $(1 + x + 2 x^{2})^{20} = a_{0} + a_{1} x + a_{2} x^{2} . . . . . a_{40} x^{40}$ , then $a_{0} + a_{2} + a_{4} . . . . a_{38}$ =

Join BYJU'S Learning Program

If (1+x+2x2)20=a0+a1x+a2x2+....a40x40. Find the values of a1+a3+a5+....a39.

If $(1 + x + 2 x^{2})^{20} = a_{0} + a_{1} x + a_{2} x^{2} + . . . . a_{40} x^{40}$ . Find the values of $a_{1} + a_{3} + a_{5} + . . . . a_{39}$ .