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 $$S_n={\displaystyle \frac{n}{2}}(a+l)$$.

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:
$$n={\displaystyle \frac{30}{2}}$$
$$n=15..........(3)$$
Hence, the total number of terms of the AP is 15.
Now, the sum of n terms of an AP is given as follows:
$$S_n={\displaystyle \frac{n}{2}}(a+l)$$
Substituting equations (1) and (3), we have:
$$S={\displaystyle \frac{15}{2}}(2+30)$$
$$S={\displaystyle \frac{15}{2}}(32)$$
$$S=15\times 16$$
$$S=240$$
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.