First slide

Binomial theorem for positive integral Index

Question

If $\sum_{r = 0}^{n} \{a_{r} (x - α + 2)^{r} - b_{r} (α - x - 1)^{r}\} = 0$ , then

Moderate

A

$b_{n} = 1 + a_{n}$
B

$b_{n} = (- 1)^{n} \times a_{n}$
C

$b_{n} = (- 1)^{n - 1} \times a_{n}$
D

$b_{n} + 1 = a_{n}$

Solution

Let $α - x - 1 = t .$
Then $\sum_{r = 0}^{n} a_{r} (1 - t)^{r} = \sum_{r = 0}^{n} b_{r} t^{r}$
$\Rightarrow b_{n} = coefficient of t^{n} in \sum_{r = 0}^{n} a_{r} (1 - t)^{r}$
$\begin{array}{l} = coefficient of t^{n} in a_{n} (1 - t)^{n} \\ = (- 1)^{n} \times a_{n} \end{array}$

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