
What is the sum of even numbers between 1 and 31?
(a) 6
(b) 28
(c) 240
(d) 512
Answer
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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 .
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:
The nth term of an AP is given as follows:
The last term of the AP is 30. Hence substituting equations (1) and (2), we have:
Simplifying, we have:
Solving for n, we get:
Hence, the total number of terms of the AP is 15.
Now, the sum of n terms of an AP is given as follows:
Substituting equations (1) and (3), we have:
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.
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:
The nth term of an AP is given as follows:
The last term of the AP is 30. Hence substituting equations (1) and (2), we have:
Simplifying, we have:
Solving for n, we get:
Hence, the total number of terms of the AP is 15.
Now, the sum of n terms of an AP is given as follows:
Substituting equations (1) and (3), we have:
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.
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