If 1-x1-x-x2 -1-x- -xn-a0-a1 x-a2 x2-a3 x3- then the value of a0-a2-a4- - - isA- 2n-B- 2n-1-2C- n-1 -2D- n-1 -2

The correct option is D (1+x)(1+x+x^2).....(1+x+.....+x^n)=a_0+a_1x+a_2x^2+a_3x^3+..... Put, x=1, 2× 3× 4× ....(n+1)=a_0+a_1+a_2+a_3+...... 0=a_0 a_1+a_2 a_3+. ...

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Byju's Answer

Standard XI

Mathematics

Binomial Expression

If 1+x1+x+x2 ...

Question

If $(1 + x) (1 + x + x^{2}) . . . . . (1 + x + . . . . . + x^{n}) = a_{0} + a_{1} x + a_{2} x^{2} + a_{3} x^{3} + . . . . .$ , then the value of $a_{0} + a_{2} + a_{4} + . . . . .$ is

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Solution

The correct option is D
$(1 + x) (1 + x + x^{2}) . . . . . (1 + x + . . . . . + x^{n}) = a_{0} + a_{1} x + a_{2} x^{2} + a_{3} x^{3} + . . . . .$
Put, $x = 1,$
$2 \times 3 \times 4 \times . . . . (n + 1) = a_{0} + a_{1} + a_{2} + a_{3} + . . . . . .$
$0 = a_{0} - a_{1} + a_{2} - a_{3} + . . . . .$
Put $x = - 1,$
$(n + 1)! = 2 [a_{0} + a_{2} + . . . .] \Rightarrow a_{0} + a_{2} + . . . . = \frac{(n + 1)!}{2}$

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Similar questions

Q. If

(1 + x) (1 + x + x^{2}) . . . . . (1 + x + . . . . . + x^{n}) = a_{0} + a_{1} x + a_{2} x^{2} + a_{3} x^{3} + . . . . .

, then the value of

a_{0} + a_{2} + a_{4} + . . . . .

Q. If

(1 + x) (1 + x + x^{2}) . . . . . (1 + x + . . . . . + x^{n}) = a_{0} + a_{1} x + a_{2} x^{2} + a_{3} x^{3} + . . . . .

, then the value of

a_{0} + a_{2} + a_{4} + . . . . .

Q. If

(1 \times x) (1 + x + x^{2}) (1 + x + x^{2} + x^{3}) . . . . . (1 + x + x^{2} + x^{3} + . . . . . . + x^{n}) \equiv

a_{0} + a_{1} x + a_{2} x^{2} + a_{3} x^{3} + . . . . . . . a_{m} x^{m}

then

m \sum r = 0 a_{r}

has the value equal to

(n + k)!

.Find

k

Q. Let

(1 + x) (1 + x + x^{2}) (1 + x + x^{2} + x^{3}) . . . . . . (1 + x + x^{2} + . . . . + x^{30}) = a_{0} + a_{1} x + a_{2} x^{2} + . . . . . . . . + a_{465} x^{465}

then sum of

a_{0} + a_{2} + a_{4} + . . . . . . . . + a_{464}

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

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If (1+x)(1+x+x2).....(1+x+.....+xn)=a0+a1x+a2x2+a3x3+....., then the value of a0+a2+a4+..... is

The correct option is D (1+x)(1+x+x2).....(1+x+.....+xn)=a0+a1x+a2x2+a3x3+..... Put, x=1, 2×3×4×....(n+1)=a0+a1+a2+a3+...... 0=a0−a1+a2−a3+..... Put x=−1, (n+1)!=2[a0+a2+....]⇒a0+a2+....=(n+1)!2

If $(1 + x) (1 + x + x^{2}) . . . . . (1 + x + . . . . . + x^{n}) = a_{0} + a_{1} x + a_{2} x^{2} + a_{3} x^{3} + . . . . .$ , then the value of $a_{0} + a_{2} + a_{4} + . . . . .$ is