First slide

Binomial theorem for positive integral Index

Question

$5$ th term from the end in the expansion of ${(\frac{x^{2}}{3} - \frac{3}{x^{2}})}^{8}$ is:

Moderate

A

$25 x^{2}$
B

$- 70$
C

$70$
D

$- 25 x^{2}$

Solution

As number of terms in the binomial expansion of ${(x + a)}^{n}$ is $(n + 1),$ the $r$ th term from the end is $((n + 1) + (- 1) (r - 1)) t h = (n + 2 - r) th$ term from the beginning.
Here $n = 8, r = 5,$ thus, $5$ th term from the end is $(8 + 2 - 5)$ th $5 t h$ term from the beginning, and it is given by
$t_{5} =^{8} C_{4} {(\frac{x^{2}}{3})}^{4} {(- \frac{3}{x^{2}})}^{8 - 4} =^{8} C_{4} = 70$

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