First slide

Binomial theorem for negative integral and rational Index

Question

If x is positive, the first negative term in the expansion of $(1 + x)^{27 / 5} is (| x | < 1)$

Moderate

A

5^th term
B

8^th term
C

6^th term
D

7^th term

Solution

$T_{r + 1} in (1 + x)^{n}$ is
$\frac{n (n - 1) (n - 2) \dots (n - r + 1)}{r!} x^{r}$
For first negative term,
$n - r + 1 < 0$
or $\frac{27}{5} - r + 1 < 0 or r > \frac{32}{5}$
Thus, first negative term occurs when r = 7.

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