Question

Find the sum of the first 123 even natural numbers.

Answer 1

Hint: Here, we are required to find the sum of the first 123 even natural numbers. Thus, we will find the sequence representing the first 123 even natural numbers. We will observe that this is an arithmetic progression and hence, we will use the formula of sum of first $n$ terms to find the required answer.

Formula Used:
Sum of $n$ terms of an AP, ${S_n} = \dfrac{n}{2}\left[ {2a + \left( {n - 1} \right)d} \right]$
Where, $a$ is the first term, $d$ is the common difference and $n$represents the total number of terms in the AP

Complete step by step solution:
As we know, natural numbers are positive integers or those numbers which are greater than equal to 1 or in other words, all positive integers except 0 are natural numbers.
Now, even natural numbers are those numbers which are divisible by 2.
Hence, the sequence of even natural numbers can be written as:
$2,4,6,8,10,....$
Here, clearly this sequence is an Arithmetic Progression because each term is greater than the preceding term by 2.
Thus, the first term, $a = 2$
Common difference, $d = 2$
And, according to the question, the total number of even natural numbers to be considered in this sequence, $n = 123$
Now, in an AP, the sum of first $n$ terms is given as:
${S_n} = \dfrac{n}{2}\left[ {2a + \left( {n - 1} \right)d} \right]$
Hence, the sum of first 123 even natural numbers is:
${S_{123}} = \dfrac{{123}}{2}\left[ {2\left( 2 \right) + \left( {123 - 1} \right)\left( 2 \right)} \right]$
$ \Rightarrow {S_{123}} = \dfrac{{123}}{2}\left[ {4 + \left( {122} \right)\left( 2 \right)} \right] = \dfrac{{123}}{2}\left[ {4 + 244} \right] = \dfrac{{123}}{2} \times 248$
Hence, solving further, we get,
$ \Rightarrow {S_{123}} = 123 \times 124$
$ \Rightarrow {S_{123}} = 15252$

Therefore, the required sum of first 123 even natural numbers is 15252
Hence, this is the required answer.

Note:
An Arithmetic Progression or A.P. is a sequence in which the difference between two consecutive terms is the same. Arithmetic progressions are also used in real life such as adding the same amount as our pocket money in our money bank. Since, we add the same amount each time, that amount or that pocket money will become our common difference in this case. Similarly, we hire a taxi, we are charged an initial rate and then rate per kilometer. That rate per kilometer becomes our common difference and each addition gives us an A.P. Hence, this is used in our day to day life.