Question

Find the $15^{th}$ term from the end for A.P. as 10, 15, 20, 25, 30, ...., 1000.

Answer 1

Hint: First, we should know the formula to calculate the $n^{th}$ term of the A.P. as ${{a}_{n}}=a+\left( n-1 \right)d$. Then, we need to find the $15^{th}$ term from the end, so we reverse the A.P as 1000, 995, 990, ....., 10.Then, by substituting the value of a as 1000 and value of d as $-5$ calculated to get the value of ${{a}_{15}}$.

Complete step-by-step answer:
In this question, we are supposed to find the $15^{th}$ term from the end for A.P. from the series given as 10, 15, 20, 25, 30, ...., 1000.
So, we should know the formula to calculate the $n^{th}$ term of the A.P. as:
${{a}_{n}}=a+\left( n-1 \right)d$
Here, in the above formula ${{a}_{n}}$ is the $n^{th}$ term of A.P where a is first term and d is the difference between the two consecutive terms.
Now, we need to find the $15^{th}$ term from the end, so we reverse the A.P as 1000, 995, 990, ....., 10.
So, a for the above given series is 1000 and to calculate d we have:
$\begin{align}
  & d=995-1000 \\
 & \Rightarrow d=-5 \\
\end{align}$
So, now to get the $15^{th}$ term from the series of A.P, we can use the formula as:
${{a}_{15}}=a+\left( 15-1 \right)d$
Now, by substituting the value of a as 1000 and value of d as $-5$ calculated above to get the value of ${{a}_{15}}$ as:
${{a}_{15}}=1000+\left( 15-1 \right)\times \left( -5 \right)$
So, now just by solving the above expression as:
 $\begin{align}
  & {{a}_{15}}=1000+\left( 14 \right)\times \left( -5 \right) \\
 & \Rightarrow {{a}_{15}}=1000+\left( -70 \right) \\
 & \Rightarrow {{a}_{15}}=930 \\
\end{align}$
So, the $15^{th}$ term from the end of the A.P is 930.
Hence, the $15^{th}$ term from the end of the A.P is 930 as the final answer.

Note: In this type of question, we must know the approach to solve as it is asked to find the $15^{th}$ term from the end and not from the beginning , that is why we reversed the entire A.P. If in a hurry we find the $15^{th}$term from the beginning we get the wrong answer as:
$\begin{align}
  & {{a}_{15}}=10+\left( 15-1 \right)\times \left( 5 \right) \\
 & \Rightarrow {{a}_{15}}=10+\left( 14 \right)\times \left( 5 \right) \\
 & \Rightarrow {{a}_{15}}=10+70 \\
 & \Rightarrow {{a}_{15}}=80 \\
\end{align}$
Which is a wrong answer and we should take care while solving the same.