Prove the following b y using the principle of mathematical induction for all $n \in N : 1 + 2 + 3 + . . . . . + n < \frac{1}{8} {(2 n + 1)}^{2}$

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Solution

Let the given statement be $P (n)$ , i.e.,
$P (n) : 1 + 2 + 3 + . . . . . + n < \frac{1}{8} {(2 n + 1)}^{2}$
$P (n)$ is true for $n = 1$ since $1 < \frac{1}{8} {(2.1)}^{2} = \frac{9}{8}$
Let $P (k)$ be true $P (n)$ be true for some $k \in N$ is true i.e.
$P (k) : 1 + 2 + 3 + . . . . . + k < \frac{1}{8} {(2 k + 1)}^{2}$ ......(i)
Consider
$(1 + 2 + . . . . . . k) + (k + 1) < \frac{1}{8} {(2 k + 1)}^{2} + (k + 1)$ [Using $(i)$ ]
$= \frac{1}{8} {{(2 k + 1)}^{2} + 8 (k + 1)}$
$= \frac{1}{8} (4 k^{2} + 4 k + 1 + 8 k + 8)$
$= \frac{1}{8} (4 k^{2} + 12 k + 9)$
$= \frac{1}{8} (2 k + 3)^{2}$
$= \frac{1}{8} {2 (k + 1) + 1}^{2}$
$(1 + 2 + 3 + . . . . . k) + (k + 1) < \frac{1}{8} {(2 k + 1)}^{2}$
Hence,
Thus, $P (k + 1)$ is true whenever $P (k)$ is true.
Hence, by the principle of mathematical induction, statement $P (n)$ is true for all natural numbers $n$

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