Question

Find the sum of even numbers between 1 and 25.
$
  {\text{A}}{\text{. 155}} \\
  {\text{B}}{\text{. 156}} \\
  {\text{C}}{\text{. 157}} \\
  {\text{D}}{\text{. 158}} \\
$

Answer 1

Hint: To determine the answer to the given question we check if the even numbers are in Arithmetic progression. Then we use the formula for sum of numbers in A.P.

Complete step-by-step answer:

Given Data,
The even numbers between 1 and 25 are 2, 4, 6……24

These numbers are in arithmetic progression.

The common difference d = 2.
(Common difference d, is the difference between any two consecutive terms in the Arithmetic Progression)
The first term a = 2.

To find the number of terms in A.P.
We use the formula, ${{\text{T}}_{\text{n}}}$= a + (n -1) d
Where ${{\text{T}}_{\text{n}}}$ is the last term and ‘a’ is the first term of the progression and n is the number of terms.

The last term (${{\text{T}}_{\text{n}}}$) = 24
⟹24 = 2 + (n-1) 2
⟹22 = (n -1) 2
⟹11 = n -1
⟹n = 12
The numbers of terms n = 12.

The sum of first ‘n’ terms of arithmetic series formula can be written as

${{\text{S}}_{\text{n}}}$ = $\dfrac{{\text{n}}}{2}$ [2a + (n -1) d]
       = $\dfrac{{12}}{2}$ [2x2 + (12 – 1) x 2]
       = 6 [4 + 11 x 2]
       = 6 [4 +22]
        = 6 x 26
${{\text{S}}_{\text{n}}}$ = 156
The sum of even numbers between 1 and 25 is 156.
Hence, Option B is the correct answer.

Note: In order to solve such type questions the key is to identify that the numbers are in Arithmetic Progression. Then apply A.P. formula to determine the answer.
An arithmetic progression or arithmetic sequence is a sequence of numbers such that the difference between the consecutive terms is constant.