Question

What is the sum of even numbers between 1 and 31?
(a) 6
(b) 28
(c) 240
(d) 512

Answer 1

Hint: The even numbers between 1 and 31 form an arithmetic progression (AP) with the first term as 2 and the last term as 30. The sum of n terms of an AP with first term a and last term l is given as ${S_n} = \dfrac{n}{2}(a + l)$.

Complete step-by-step answer:
We need to find the sum of the even numbers between 1 and 31.
The even numbers between 1 and 31 are given as follows:
2, 4, 6, …, 30
Arithmetic progression (AP) is a sequence of numbers in which each differs from the preceding one by a constant quantity. This constant number is called the common difference.
The even numbers between 1 and 31 also form an AP with the first term as 2 and the common difference is 2. Hence, we have:
$a = 2............(1)$
$d = 2...........(2)$
The nth term of an AP is given as follows:
${t_n} = a + (n - 1)d$
The last term of the AP is 30. Hence substituting equations (1) and (2), we have:
$30 = 2 + (n - 1)2$
Simplifying, we have:
$30 = 2 + 2n - 2$
$30 = 2n$
Solving for n, we get:
$n = \dfrac{{30}}{2}$
$n = 15..........(3)$
Hence, the total number of terms of the AP is 15.
Now, the sum of n terms of an AP is given as follows:
${S_n} = \dfrac{n}{2}(a + l)$
Substituting equations (1) and (3), we have:
$S = \dfrac{{15}}{2}(2 + 30)$
$S = \dfrac{{15}}{2}(32)$
$S = 15 \times 16$
$S = 240$
Hence, the correct answer is option (c).

Note: You can also add the even numbers directly but it is time-consuming and hence, this method is a shortcut to find the sum of even numbers between 1 and 31.