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Permutations and Combinations

If ∑r=1k r = ...

Question

If $k \sum r = 1 r = \frac{1}{2} (n^{2} + 11 n + 30)$ is true for $n \in N,$ then $k =$
(use principle of mathematical induction)

A

n + 7

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B

n + 6

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C

n + 4

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D

n + 5

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Solution

The correct option is D $n + 5$
$\frac{1}{2} (n^{2} + 11 n + 30) = \frac{1}{2} (n + 5) (n + 6)$
Here, $1 + 2 + 3 + \dots + k = \frac{k (k + 1)}{2}$
[from mathematical induction]
By replacing $k$ by $n + 5$ ,
$∴ 1 + 2 + 3 + \dots (n + 4) + (n + 5) = \frac{(n + 5) (n + 6)}{2}$
$∴ k = (n + 5)$

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Permutations and Combinations

Standard XII Mathematics

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