Question

What is the sum of first 20 odd natural numbers?
A) $100$
B) $210$
C) $400$
D) $420$

Answer 1

Hint: In this question, we have to find the sum of first $20$ odd natural numbers. First, find the first 20 odd numbers. To find the $20^{th}$ odd number, use the A.P formula of general term. After you have all the terms, use the formula of sum of A.P to find the sum. Put all the values and simplify. You will get your answer.

Formula used: 1) ${a_n} = a + \left( {n - 1} \right)d$
2) ${S_n} = \dfrac{n}{2}\left( {2a + \left( {n - 1} \right)d} \right)$ or ${S_n} = \dfrac{n}{2}\left( {a + {a_n}} \right)$

Complete step-by-step solution:
We have to find the sum of the first $20$ odd natural numbers. So, first we will find the first $20$ odd natural numbers. We know that the first few odd numbers are $1,3,5,7,...$ but we don’t know which is the last number? So, we will find the last and the ${20^{th}}$ odd numbers using the A.P formula of general term.
Finding the ${20^{th}}$ term,
$ \Rightarrow {a_{20}} = a + \left( {n - 1} \right)d$
Putting $a = 1,$ $n = 20,$ $d = 2$,
$ \Rightarrow {a_{20}} = 1 + \left( {20 - 1} \right)2$
Solving to find the answer,
$ \Rightarrow {a_{20}} = 1 + 19 \times 2$
$ \Rightarrow {a_{20}} = 39$
Hence, ${20^{th}}$ odd number is $39$.
Next, we will find the sum of these first $20$ odd natural numbers. We will use the formula ${S_n} = \dfrac{n}{2}\left( {a + {a_n}} \right)$ to find the required sum.
Put $n = 20,$$a$ is the first term. So, $a = 1$ and ${a_n}$ is the last term. ${a_n} = 39$.
Putting all the values in the formula –
$ \Rightarrow {S_{20}} = \dfrac{{20}}{2}\left( {1 + 39} \right)$
Simplifying to find the sum,
$ \Rightarrow {S_{20}} = 10 \times 40 = 400$

$\therefore $ The sum of first $20$ odd natural numbers is option C)$400$.

Note: There is one more formula to find the sum of A.P. $ \Rightarrow {S_n} = \dfrac{n}{2}\left( {2a + \left( {n - 1} \right)d} \right)$. Using this formula, you need not even find the ${20^{th}}$ term first. You can directly put all the information that you have about the first term, common difference and the number of terms whose sum is to be found.
Using this formula to find the answer to the above question,
$ \Rightarrow {S_{20}} = \dfrac{{20}}{2}\left( {2 \times 1 + \left( {20 - 1} \right)2} \right)$
Simplifying to solve further,
$ \Rightarrow {S_{20}} = 10 \times \left( {2 + 38} \right)$
$ \Rightarrow {S_{20}} = 10 \times 40 = 400$
Therefore, the sum of the first $20$ odd natural numbers is $400$.