If The Sum of The 4th Term and The 12th Term of an Arithmetic Progress, What is the Sum of the First 15 terms of the Progression?

Question: If the sum of the 4th term and the 12th term of an arithmetic progression is 8, what is the sum of the first 15 terms of the progression?

1. 60
2. 120
3. 160
4. 240
5. 840

Solution and Explanation: 

Approach Solution 1:

The formula for the nth Term of an Arithmetic Progression is Tn =a+(n−1)d
The formula for the Sum of n terms is Sn =(n/2)∗[2a+(n−1)∗d
Where, a = first term of Progression, d = common difference (Second term - first term or Third - second term etc.)
As per the question the sum of the 4th term and the 12th term is 8, hence as per the formula of 4th term and 12th term,
4th Term is T4 =a+(4−1)∗d=a+3d
12th Term is T12=a+(12−1)∗d=a+11d

Given the above equation where the sum is 8, the formula for that should be:
(a+11d) + (a+3d) = 8
i.e. 2a + 14d = 8

As per the question, we need to find the sum of 15 terms, 
Hence, as per the arithmetic progression,

S15=(15/2)∗[2a+(15−1)∗d
=(15/2)∗[2a+14d]
=(15/2)∗8
= 7.5*8
= 60
Hence, the correct answer is option A.

Approach Solution 2:

This problem can be solved by considering the (nth) term for an Arithmetic Progression.
Let us consider the nth term of an Arithmetic Progression (AP)
The formula considering nth term of AP is a+(n-1)d

Considering the above formula, the
1st term of AP = a+d
And the 12th term of AP = a+11d

As per the question, the sum of 1st term and 12th term is 8
Hence, if we add those, the formula becomes:
a+d+a+11d =8
2a+14d=8
a+7d = 4 ---(i)

Now, to find the sum of the 1st 15 terms =
(a+d)+(a+2d)+(a+3d)+........(a+14d)
=15a+(d+2d+3d+4d+..14d)
= 15a+d(1+2+3+..14)

Sum of 1st n natural numbers = n(n+1) /2
= 15a +d (14∗15/2)
= 15a +d(7*15)
=15(a+7d) 
As per the equation (i), the value of a+7d is 4. Considering 4 in place of (a+7d)
=15*4=60
Hence, the correct answer is A.

Approach Solution 3:

We know that 4th term + 12th term = 8
i.e., (a+3d)+(a+11d) = 8
2a+14d

= 8 ---- Let us consider this as equation (1)

Now, let us take the sum of the first 15 terms

= (15/2) * [2a + (15-1)d]
= (15/2) * [2a + 14d]

Entering the value of (2a + 14d) from equation 1
= (15/2) * 8 
= 60

Hence, A is the correct answer.

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