If a and b denote the sum of the coefficients of $x^{n}$ in the expansions of $(1 - 3 x + 10 x^{2})^{n}$ and $(1 + x^{2})^{n}$ respectively, then write the relation between a and b.

Open in App

Solution

$(1 - 3 x + 10 x^{2})^{n}$

$=^{n} C_{0} (1)^{n} +^{n} C_{1} (1)^{n - 1}$

$(- 3 x + 10 x^{2}) + . . . +^{n} C_{n} (- 3 x + 10 x^{2})^{2}$

Let x =1 on both sides.

$(1 - 3 + 10)^{n}$

$=^{n} C_{0} (1)^{n} +^{n} C_{1} (1)^{n - 1} (- 3 + 10)^{n}$

$∴ 8^{n} = a$

$(1 + x^{2})^{n} =^{n} C - 0 (1)^{n} +^{n} C_{1} (1)^{n - 1} (x^{2}) + . . . . +^{n} C_{n} (x^{2})^{n}$

Let x =1 on both sides.

$(1 + 1)^{n}$

$=^{n} C_{0} (1)^{n} +^{n} C_{1} (1)^{n - 1} (1) + . . . . +^{n} C_{n} (1)^{n}$

$∴ 2^{n} = b$

$∴ a = b^{3}$

Suggest Corrections

Similar questions

{(1 + x)}^{2 n} and {(1 + x)}^{2 n - 1}

x^{n}

(1 + x)^{2 n}

(1 + x)^{2 n - 1}

(1 - 3 x + 10 x^{2})^{n}

a

(1 + x^{2})^{n}

(1 - 3 x + 10 x^{2})^{n}

(1 + x^{2})^{n}

(1 - 3 x + 10 x^{2})^{n}

a

(1 + x^{2})^{n}

b

Join BYJU'S Learning Program