Question

Prove that by the method of induction; for all $n \in \mathbb{N},3 + 7 + 11 + .......... + {n^{th}}term = n(2n + 1)$ .

Answer 1

Hint: First prove that it is true for $n = 1$ then assume that it is true for $n = k$ now use the result of $n = k$ to prove that it is true for $n = k + 1$ for all $n \in \mathbb{N}$ .
Complete step by step answer:
Let’s see weather it is true for $n = 1$ Therefore we will check for the ${1}^{st}$ term only
Here in LHS first term is 3
In RHS, we will get it as

n(2n + 1)\\\ = 1(2 \times 1 + 1)\\\ = 2 + 1\\\ = 3 \end{array}$$ Hence LHS=RHS Now for $$n = k$$ , We will get it as $$3 + 7 + 11 + .......... + {k^{th}}term = k(2k + 1)$$ Assuming that it is true for $$n = k$$ Let us consider $$n = k + 1$$ Therefore LHS becomes $$3 + 7 + 11 + ......... + {k^{th}}term + {(k + 1)^{th}}term$$ Now let us try to find a general term for the given series 3,7,11,...... So the number is increasing by 4 each time and the first number is 3 in the series therefore we can write the general form as $$4k + 3,\forall k = 0,1,2,3,.........n$$ So when $$k = 0$$ we are getting 3 when $$k = 1$$ we are getting 7 So similarly the $$k + {1^{th}}term$$ will be $$4k + 3$$ Putting this in the place of $$k + {1^{th}}term$$ and $$k(2k + 1)$$ for all the terms before that as before that the terms where till $${k^{th}}term$$ and we already assumed that $$k(2k + 1)$$ is true upto there. So now the LHS becomes $$\begin{array}{l} k(2k + 1) + 4k + 3\\\ = 2{k^2} + k + 4k + 3\\\ = 2{k^2} + 5k + 3\\\ = (k + 1)(2k + 3)\\\ = (k + 1)\\{ 2(k + 1) + 1\\} \end{array}$$ Now let us compare with RHS by putting k+1 in $$n(2n + 1)$$ we will get it as $$(k + 1)\\{ 2(k + 1) + 1\\} $$ Therefore LHS=RHS (Hence Proved) Note: By the principle of mathematical induction we can always always assume something to be true and apply it to prove another. Given that the range of both must be the same i.e., in this case we were dealing with natural numbers so we cannot assume one thing in natural numbers and put it on a different spectrum like rational or real numbers.