First slide

Binomial theorem for positive integral Index

Question

1.2.3 + 2.3.4 + 3.4.5 + … upto n terms is equal to

Moderate

A

$\frac{1}{4} (n + 1) (n + 2) (n + 3)$
B

$\frac{1}{4} (n + 2) (n - 2) (n - 3) (n + 3)$
C

$\frac{1}{4} n (n + 1) (n + 2) (n + 3)$
D

none of these

Solution

$\begin{array}{l} t_{r} = r (r + 1) (r + 2) = (r + 1) \{(r + 1)^{2} - 1\} \\ = (r + 1)^{3} - (r + 1) \\ \Rightarrow \sum_{r = 1}^{n} t_{r} = \sum_{r = 1}^{n + 1} r^{3} - \sum_{r = 1}^{n + 1} r \\ = \frac{1}{4} (n + 1)^{2} (n + 2)^{2} - \frac{1}{2} (n + 2) (n + 1) \\ = \frac{1}{4} (n + 1) (n + 2) (n^{2} + 3 n + 2 - 2) \\ = \frac{1}{4} n (n + 1) (n + 2) (n + 3) . \end{array}$

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