Question

If the \[{8^{th}}\] of an A.P. is 37 and the \[{15^{th}}\] is 15 more than the \[{12^{th}}\] term, find the A.P. Also find the sum of the first 20 terms of this A.P.

Answer 1

Hint: The general term of an AP is given by \[{a_n} = a + (n - 1)d\] where \[{a_n}\] is the \[{n^{th}}\] term and a is the first term of the AP which makes d the common difference between 2 terms of an AP. The sum of n terms is given by \[{S_n} = \dfrac{n}{2}\left\{ {2a + (n - 1)d} \right\}\] where \[{S_n}\] is the sum of AP till \[{n^{th}}\] term.

Complete step-by-step answer:
 With the given formulas in hint let us try to formulate some equations which will ultimately give us the answers.
It is given that the \[{8^{th}}\] term is 37
\[\begin{array}{l}
\therefore {a_8} = 37 = a + (8 - 1)d\\
 \Rightarrow a + 7d = 37.....................................(i)
\end{array}\]
Now using the relation of \[{15^{th}}\] and \[{12^{th}}\] we get an equation
\[\begin{array}{l}
\therefore {a_{15}} = {a_{12}} + 15\\
 \Rightarrow a + 14d = a + 11d + 15\\
 \Rightarrow 14d - 11d = a - a + 15\\
 \Rightarrow 3d = 15\\
 \Rightarrow d = 5.............................................(ii)
\end{array}\]
Now as we have the value of d let us put it in equation (i) to get the value of a.
\[\begin{array}{l}
\therefore a + 7d = 37\\
 \Rightarrow a + 7 \times 5 = 37\\
 \Rightarrow a = 37 - 35\\
 \Rightarrow a = 2
\end{array}\]
So now as we have the value of a and d it's easy to get the AP which is
 \[2,7,12,17,22,27,32,37,42.........\]
Now as we have the AP and all the other values let us try to find the sum of first 20 terms
We know that \[{S_n} = \dfrac{n}{2}\left\{ {2a + (n - 1)d} \right\}\]
Therefore if we substitute \[a = 2,n = 20\& d = 5\] we will get it as
\[\begin{array}{l}
\therefore {S_{20}} = \dfrac{{20}}{2}\left\{ {2 \times 2 + (20 - 1) \times 5} \right\}\\
 \Rightarrow {S_{20}} = 10 \times \{ 4 + 19 \times 5\} \\
 \Rightarrow {S_{20}} = 10 \times \{ 4 + 95\} \\
 \Rightarrow {S_{20}} = 10 \times 99\\
 \Rightarrow {S_{20}} = 990
\end{array}\]

Note: Students often make mistake while formulating or solving the equations beware of the signs as they can be the reason for incorrect answer and also use the formula to get the sum of AP dont try to write till the \[{20^{th}}\] and then add them all that will consume a lot of time and indeed the chances of making mistakes are higher.