First slide

Binomial theorem for positive integral Index

Question

In the expansion of (a + b)ⁿ, if two consecutive terms are equal, then which of the following is/are always integer?

Moderate

A

$\frac{(n + 1) b}{a + b}$
B

$\frac{(n + 1) a}{a + b}$
C

$\frac{na}{a - b}$
D

$\frac{na}{a + b}$

Solution

We have (a + b)ⁿ
$\frac{T_{r + 1}}{T_{r}} = \frac{n - r + 1}{r} \cdot \frac{b}{a} = 1$
$∴ (n + 1) b - br = ar$
$∴ r = \frac{(n + 1) b}{a + b} is integer$
Also, considering (b + a)ⁿ.
$\frac{T_{r + 1}}{T_{r}} = \frac{n - r + 1}{r} \cdot \frac{a}{b} = 1$
$∴ r = \frac{(n + 1) a}{a + b} is integer$

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