QUESTION

Prove that $\left( {2n + 7} \right) < {\left( {n + 3} \right)^2}$.

Hint: In order to prove this statement, we must assume n as any natural number. We should try to solve this question by the principle of mathematical induction.

Mathematical induction is used to prove a statement or a theorem which is true for every natural number. In other words, we can also use this mathematical induction principle to prove the proportional relationship between the natural numbers.

Let us assume any natural number n

So, we will take n=1 in the given condition-
$\left( {2n + 7} \right) < {\left( {n + 3} \right)^2}$
$\Rightarrow$ $\left( {2 \times 1 + 7} \right) < {\left( {1 + 3} \right)^2}$
$\Rightarrow$ $\left( {2 + 7} \right) < {\left( 4 \right)^2}$
$\Rightarrow$ $\left( 9 \right) < \left( {16} \right)$

$\therefore$ we conclude that for any natural number, this given condition would be satisfied.

Note: To solve this type of question we can also use other natural numbers. Another way to solve this question is we can expand (n+3)$^2$ and then put any natural number in place of n. But the above mentioned way is most appropriate to solve this type of question.