Question

Given the first and last term, a = 6, \[{{t}_{9}}\] = 24, how do you find the sum of the arithmetic series?

Answer 1

Hint: To determine the sum of an arithmetic series, we can use formula and proceed with the solution. Here the initial term is given as 6 and the last term is given as 24. So we have a formula to find the sum of the arithmetic series using first term and last term using the formulas.

Complete step by step solution:
Here we have,
First term, a = 6
Last term, l = 24
It is mentioned in question that $9^{th}$ term is the last term in the given arithmetic series,
So, n = 9
The sum of the given arithmetic series using first term and last term is
\[{{S}_{n}}=\dfrac{n}{2}\left( a+l \right)\]
Let’s substitute the values provided in the question,
\[\Rightarrow {{S}_{9}}=\dfrac{9}{2}\left( 6+24 \right)\]
Add the terms inside the bracket,
\[\Rightarrow {{S}_{9}}=\dfrac{9}{2}\left( 30 \right)\]
Divide 30 on the numerator by 2 to obtain 15,
\[\Rightarrow {{S}_{9}}=9\times 15\]
Multiply 9 and 15 to obtain the final answer to get the sum of the arithmetic series,
\[\Rightarrow {{S}_{9}}=135\]

So the sum of the given arithmetic series is given by 135.

Note: It is important to remember the formula of finding the sum of the arithmetic series when the last term and first term of the arithmetic series. Student can go wrong in finding the sum if he makes any mistake in writing the formula. Be careful in solving the arithmetic operation after substitution of given values in the formula.