Question

Find the ${{15}^{th}}$ term of the AP $-40,-15,10,35,\ldots \ldots $.

Answer 1

Hint: In this question, we are given an arithmetic progression and we need to find its ${{15}^{th}}$ term which means we need to find ${{a}_{15}}$ for the given arithmetic progression. For this, we will first find the first term 'a' of the arithmetic progression. After that, we will make the difference between the second term and the first term or the difference between the third term and the second term to calculate the common difference 'd'. Using these values in the formula ${{a}_{n}}=a+\left( n-1 \right)d$ (Where n is the term which we want to find) we will find the ${{15}^{th}}$ term.

Complete step by step answer:
Here we are given the arithmetic progression as $-40,-15,10,35,\ldots \ldots $.
We can see that the first term of the arithmetic progression is -40, so we can say that the value of a is -40. Now let us calculate the common difference of the arithmetic progression. For this, let us make the difference between the second term and the first term, we get $\left( -15 \right)-\left( -40 \right)=-15+40=25$.
Hence the common difference of the given arithmetic progression is 25, so we can say d = 25.
We know that, ${{n}^{th}}$ term of an arithmetic progression can be found using the formula ${{a}_{n}}=a+\left( n-1 \right)d$.
Here we need to find the ${{15}^{th}}$ term which means we have to put n = 15.
Putting in the values of a, d and n in the formula, we get:
$\begin{align}
  & {{a}_{15}}=-40+\left( 15-1 \right)\left( 25 \right) \\
 & \Rightarrow {{a}_{15}}=-40+\left( 14 \right)\left( 25 \right) \\
\end{align}$
Multiplying 25 by 14 gives us 350 so we get:
$\begin{align}
  & {{a}_{15}}=-40+350 \\
 & \Rightarrow {{a}_{15}}=310 \\
\end{align}$
Hence the ${{15}^{th}}$ term of the given arithmetic progression $-40,-15,10,35,\ldots \ldots $ is equal to 310.

Note:
Students should take care of the signs while solving the equation. They should note that the common difference d and the first term a can be negative too. Students often get confused between the terms ${{a}_{n}}$ and n. ${{a}_{n}}$ is the ${{n}^{th}}$ term of the arithmetic progression. For finding the common difference, students can also subtract the second term from the third term or the third term from the fourth term.