Question

The sum of first 45 natural numbers is:
(a) 1035
(b) 1280
(c) 2070
(d) 2140

Answer 1

Hint: We can use here the concept that whenever the difference between successive consecutive terms of a sequence is constant, it forms an AP and thereafter we will use the formula to find the sum of n terms of an AP to get the sum of first 45 natural numbers.

Complete step-by-step answer:
The first 45 natural numbers can be written as:

1, 2, 3 , 3, ……. , 45.

Here, we can see that the difference between any two successive consecutive terms of this series is 1 which is a constant, so this sequence forms an AP.

The constant 1 which is the difference of two terms here is called the common difference of the AP.

So, we have an AP with the first term “a”= 1 and also the common difference “d” = 1.

Now, to find the sum of this series we will use the formula for the sum of n terms of an AP,

which is given as:

${{S}_{n}}=\dfrac{n}{2}\left( 2a+\left( n-1 \right)d \right)$

Since, in the given series we have 45 terms, so we have the value of n equal to 45, a = 1 and also d = 1. So, we will substitute these values in this formula to get the sum:

$\begin{align}
  & {{S}_{45}}=\dfrac{45}{2}\left( 2\times 1+\left( 45-1 \right)1 \right) \\
 & \,\,\,\,\,\,\,\,=\dfrac{45}{2}\left( 2+44 \right) \\
 & \,\,\,\,\,\,\,\,=\dfrac{45}{2}\times 46 \\
 & \,\,\,\,\,\,\,\,=45\times 23 \\
 & \,\,\,\,\,\,\,\,=1035 \\
\end{align}$

Therefore, the value of the sum of the first 45 natural numbers is 1035.

Hence, option (a) is the correct answer.

Note: It should be noted here that we can also use a shortcut formula used to find the sum of n terms of an AP, which is ${{S}_{n}}=\dfrac{n}{2}\left( first\,\,term+last\,\,term \right)$. Since, here both the first term and the last term are known, using this formula may save our time.