Question

Find the sum of all even numbers from 1 to 350.

Answer 1

Hint: Approach these types of questions with the help of Arithmetic progression. Use the formula ${a_n} = a + (n - 1)d$to find the number of terms n. Use the concept of Arithmetic Progression in this question which is known as a sequence of numbers in which the difference of any two adjacent terms is constant. To find out Sum of first n term of an AP, use the formula ${S_n} = \dfrac{n}{2}[2a + (n - 1)d]$

Complete step by step answer:
As per the question, we need to find the sum of all the even numbers that is
2, 4, 6, ……..350
So in this question a series is made which is nothing but a Arithmetic progression
Where first term a = 2
Common difference d = 2
Last Term ${a_n}$ = 350
Therefore, as we know the last term is given by ${a_n} = a + (n - 1)d$
Now, we will put all the values into the formula, for finding out the term, so we have
350 = 2 + (n – 1)2
350 = 2 + 2n – 2
350 = 2n
n = $\dfrac{{350}}{2}$
From above we obtain the number of terms which is n = 175
For finding out the sum of all the even numbers, we will use the formula of Sum of First Terms of an Arithmetic Progression, that is ${S_n} = \dfrac{n}{2}[2a + (n - 1)d]$
So, we have
n = 175
a = 2
d = 2
Now put the values in to the formula,
${S_{175}} = \dfrac{{175}}{2}[2(2) + (175 - 1)(2)]$
${S_{175}} = 30800$

So, the value of sum of all the even numbers is 30800

Note: Another method:
To find the sum we have direct formula too i.e., ${S_n} = \dfrac{n}{2}(a + {a_n})$
Where ${a_n}$ is the last term
Now from the question
${a_n}$ = 350
a = 2
n = 175
Then from the formula,
${S_{175}} = \dfrac{{175}}{2}(2 + 350)$
${S_{175}} = \dfrac{{175}}{2}(352)$
${S_{175}} = 30800$