Name: Sum of Infinite Terms of G.P.
Uploaded: 2019-07-19
Duration: 3 min 36 s
Description: Do you know that we can find the sum of an infinite number of terms of G.P.? Of course, there will be some conditions. Check this video to learn more.

Question

The first four terms, in ascending powers of x, in the expansion of ${(1 + ax + {bx}^{2})}^{7}$ , are $1 - 14 x + 91 x^{2} + {px}^{3}$ . Find the values of $a, b$ and $p$ .

Accepted Answer

Given that: The first four terms, in ascending powers of x, in the expansion of ${(1 + ax + {bx}^{2})}^{7}$ , are $1 - 14 x + 91 x^{2} + {px}^{3}$

We have to find the values of $a, b$ and $p$ .

We have, ${(1 + ax + {bx}^{2})}^{7}$

As we know that,

${(a + b)}^{n} = (\begin{matrix} n \\ 0 \end{matrix}) a^{n} + (\begin{matrix} n \\ 1 \end{matrix}) a^{n - 1} b^{1} + (\begin{matrix} n \\ 2 \end{matrix}) a^{n - 2} b^{2} . . . . . (\begin{matrix} n \\ n \end{matrix}) b$

So, ${(1 + ax + {bx}^{2})}^{7} = [(\begin{matrix} 7 \\ 0 \end{matrix}) {(1)}^{7} + (\begin{matrix} 7 \\ 1 \end{matrix}) {(1)}^{6} (ax + {bx}^{2}) + (\begin{matrix} 7 \\ 2 \end{matrix}) {(1)}^{5} {(ax + {bx}^{2})}^{2} . . . . .]$

$\Rightarrow {(1 + ax + {bx}^{2})}^{7} = [1 {(1)}^{7} + 7 {(1)}^{6} (ax + {bx}^{2}) + 21 {(1)}^{5} {(ax + {bx}^{2})}^{2} . . . . .]$

$\Rightarrow {(1 + ax + {bx}^{2})}^{7} = 1 + 7 a x + 91 b x^{2} - 364 x^{3}$

But, ${(1 + ax + {bx}^{2})}^{7} = 1 - 14 x + 91 x^{2} + p x^{3}$ (given)

So, the required value of $a = - 2, b = 1 p = - 364$

The first four terms, in ascending powers of x, in the expansion of ${(1 + ax + {bx}^{2})}^{7}$ , are $1 - 14 x + 91 x^{2} + {px}^{3}$ . Find the values of $a, b$ and $p$ .

Important Questions on Series

Learn and Prepare for any exam you want !

The first four terms, in ascending powers of x, in the expansion of 1+ax+bx27, are 1-14x+91x2+px3. Find the values of a, b and p.

Important Questions on Series

Learn and Prepare for any exam you want !

The first four terms, in ascending powers of x, in the expansion of ${(1 + ax + {bx}^{2})}^{7}$ , are $1 - 14 x + 91 x^{2} + {px}^{3}$ . Find the values of $a, b$ and $p$ .