Name: The Binomial for a Negative and Fractional Index
Uploaded: 2019-07-19
Duration: 5 min 16 s
Description: This video discussed the binomial theorem for a negative and fractional index, how we can use the binomial theorem for any rational number. Must watch it.

Question

Find the two middle terms in the expansion of

{(3 x - \frac{2}{x^{2}})}^{15}

.

Accepted Answer

Given, ${(3 x - \frac{2}{x^{2}})}^{15}$

When $n$ is odd, there are two middle terms i.e., ${(\frac{n + 1}{2})}^{t h}$ and ${(\frac{n + 3}{2})}^{t h}$ terms

Therefore, middle terms in the given expansion $= {(\frac{15 + 1}{2})}^{t h}$ and ${(\frac{15 + 3}{2})}^{t h}$ terms i.e., $8^{t h}$ and $9^{t h}$ terms

General term in the expansion of ${(a + b)}^{n}$ is $T_{r + 1} = {}^{n}C_{r} a^{n - r} b^{r}$

$8^{t h}$ term in the expansion of ${(3 x - \frac{2}{x^{2}})}^{15}$ is $T_{8} = {}^{15}C_{7} {(3 x)}^{15 - 7} {(\frac{- 2}{x^{2}})}^{7}$

$= - \frac{15 \times 14 \times 13 \times 12 \times 11 \times 10 \times 9}{7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1} (3^{8} x^{8}) (\frac{2^{7}}{x^{14}})$

$= - \frac{5 \times 13 \times 11 \times 9 \times 3^{8} \times 2^{7}}{x^{6}}$

$\Rightarrow T_{8} = - \frac{6435 \times 3^{8} \times 2^{7}}{x^{6}}$

$9^{t h}$ term in the expansion of ${(3 x - \frac{2}{x^{2}})}^{15}$ is $T_{9} = {}^{15}C_{8} {(3 x)}^{15 - 8} {(\frac{- 2}{x^{2}})}^{8}$

$= - \frac{15 \times 14 \times 13 \times 12 \times 11 \times 10 \times 9}{7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1} (3^{7} x^{7}) (\frac{2^{8}}{x^{16}})$ $[∵ {}^{n}C_{r} = {}^{n}C_{n - r}]$

$= - \frac{5 \times 13 \times 11 \times 9 \times 3^{7} \times 2^{8}}{x^{9}}$

$\Rightarrow T_{9} = - \frac{6435 \times 3^{7} \times 2^{8}}{x^{9}}$

Therefore, two middle terms in the expansion of ${(3 x - \frac{2}{x^{2}})}^{15}$ are $- \frac{6435 \times 3^{8} \times 2^{7}}{x^{6}}$ and $- \frac{6435 \times 3^{7} \times 2^{8}}{x^{9}}$ .

Find the two middle terms in the expansion of ${(3 x - \frac{2}{x^{2}})}^{15}$ .

Important Questions on Binomial Theorem

Learn and Prepare for any exam you want !

Find the two middle terms in the expansion of 3x-2x215.

Important Questions on Binomial Theorem

Learn and Prepare for any exam you want !

Find the two middle terms in the expansion of ${(3 x - \frac{2}{x^{2}})}^{15}$ .