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# The sum of first $45$ natural numbers is:A. $1035$B. $1075$C. $2045$D. $2076$

Last updated date: 25th Mar 2023
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Hint: Use the general formula for the sum of the first n natural numbers, for any $\text{n }\in \text{ }\mathbb{N}$. The general formula is given by $\dfrac{\text{n(n+1)}}{\text{2}}$. Put n = 45 to find out the answer.

We know that the sum of first n natural numbers = $\dfrac{\text{n(n+1)}}{\text{2}}$ .....(i)
Here, we have to find the sum of the first 45 natural numbers. The above formula is to be used to find the correct answer.
Now, putting n = 45 in the formula (i), we get,
Sum of first 45 natural numbers
\begin{align} & =\text{ }\dfrac{45(45+1)}{2} \\ & =\text{ 45 x 23} \\ & \text{= 1035} \\ \end{align}
Thus, the correct answer is option A.

Note: This problem can be solved alternatively in a much simpler way without the use of formula.
Let us define, $\text{S = 1 + 2 + 3 +}.........\text{+ 45}$ ...... (A)
Thus, S is the sum of the first 45 natural numbers.
Arranging (A) in the reverse way, we get,
$\text{S = 45 + 44 + 43 +}.......\text{+ 1}$ ...... (B)
Observe that, there are 45 terms in each of the expressions (A) and (B). Thus, by adding (A) and (B), we get
\begin{align} & \text{2S = 46 +46 +46 +}......\text{+ 46} \\ & \Rightarrow \text{ 2S = 45 x 46} \\ & \Rightarrow \text{ S = }\dfrac{45\text{ x 46}}{2} \\ & \Rightarrow \text{ S = 45 x 23} \\ & \therefore \text{ S = 1035} \\ \end{align}
Thus, solving in the alternative way as well, we get the correct answer as option A.