Sum of an A.P.

Fill in available details into the summation formula

The sum of the first 'n' terms of an arithmetic progression = \\frac{n}{2}[2t_1 + [n - 1]*d] \\)

∴ Sum of the first 15 terms of an AP = \\frac{15}{2}[2t_1 + [15 - 1]*d] \\) = \\frac{15}{2}[2t_1 + 14d] \\)

We can find the answer either if we know t₁ and d or if we can find the value of [2t₁ + 14d].

Find the missing information from the data given and arrive at the answer

The sum of the 4^th and 12^th term = 20.

Let t₁ be the first term, t₄ be the 4^th term, and t₁₂ be the 12^th term of the progression.

Then t₄ + t₁₂ = 20

t₄ can be expressed as t₁ + 3d

Similarly, t₁₂ can be expressed as t₁ + 11d

∴ t₄ + t₁₂ = 20 can be expressed as   t₁ + 3d + t₁ + 11d = 20

2t₁ + 14d = 20

Sum of the first 15 terms = \\frac{15}{2}[2t_1 + 14d] \\)

Substitute 2t₁ + 14d = 20 in the above expression.

Sum = \\frac{15}{2} * 20 \\) = 150.

Choice C is the correct answer.

Sum of an AP = middle term * number of terms

The middle term of an arithmetic sequence of 15 terms is the 8th term.

So, sum of first 15 terms = t₈ * 15

The 8th term t₈ = \\frac{t_4 + t_{12}}{2} = \frac{20}{2} \\) = 10

Hence, sum of first 15 terms = 10 * 15 = 150

Choice C is the correct answer.