# The sum of first $45$ natural numbers is:

A. $1035$

B. $1075$

C. $2045$

D. $2076$

Answer

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362.4k+ views

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.

Complete step-by-step 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.

Complete step-by-step 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.

Last updated date: 03rd Oct 2023

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