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.