Question

The sum of the fourth and twelfth term of an arithmetic progression is 20. What is the sum of the first 15 terms of the arithmetic progression?

Answer 1

Use this formula..
http://www.google.com/imgres?imgurl=https://upload.wikimedia.org/math/a/f/e/afe20f89d7bfdbd0a191168d80eb8077.png&imgrefurl=https://en.wikipedia.org/wiki/Arithmetic_progression&h=36&w=143&tbnid=aBxBr_OUTHOXVM:&tbnh=34&tbnw=138&usg=___c442AeI6doDxq7FMLH2mdyKGMM=&docid=pxJjQA_UHEHWxM&sa=X&ved=0CCAQ9QEwAGoVChMIlfjerObJyAIVSns-Ch3fnQRI

Answer 2

we can write these formulas:
\[\begin{gathered}
{a_4} = {a_1} + 3d \hfill \\
{a_{12}} = {a_1} + 11d \hfill \\
\end{gathered} \]
where \(a_1,a_4,a_{12}\) are the first, the fourth and the twelft terms of your progression, and \(d\) is the constant of progression

Answer 3

using your condtition we get:
\[{a_4} + {a_{12}} = 2{a_1} + 14d = 20\]
or:
\[{a_1} + 7d = 10\]

Answer 4

finally we can write:
\[\begin{gathered}
{S_{15}} = \frac{{{a_1} + {a_{15}}}}{2} \cdot 15 = \frac{{{a_1} + {a_1} + 14d}}{2} \cdot 15 = \hfill \\
\hfill \\
= \frac{{2{a_1} + 14d}}{2} \cdot 15 = \left( {{a_1} + 7d} \right) \cdot 15 = 10 \cdot 15 = ...? \hfill \\
\end{gathered} \]