Question

Find the 12th term from the end of the following arithmetic progression.
3, 5, 7, 9 ……… 201.

Answer 1

Hint: In such questions in which we are asked to get the terms from the back, we first find the total number of terms in the arithmetic progression.
The formula for writing the \[{{n}^{th}}\] term of an arithmetic progression is as follows
\[{{a}_{n}}=a+(n-1)d\]
(Where a is the first term of the arithmetic progression and d is the common difference for the arithmetic progression)

Complete step-by-step answer:
As mentioned in the question, we have to find the 12th term from the end of the given arithmetic progression.
Now, we can write the final term of this given arithmetic progression as follows
\[\begin{align}
  & {{a}_{n}}=3+(n-1)2 \\
 & 201=3+2n-2 \\
 & 201=2n+1 \\
 & n=\dfrac{200}{2} \\
 & n=100 \\
\end{align}\]
(As the first term of the given arithmetic progression is 3 and the common difference for the arithmetic progression is 2)
Hence, the total number of terms of the arithmetic progression is 100.
Now, we can reverse the given arithmetic progression and change the common difference from 2 to -2 and we can take the first term to be 201.
Now, for the 12th term from the end, we can write as follows
 \[\begin{align}
  & {{a}_{12}}=201+(12-1)(-2) \\
 & {{a}_{12}}=201-22=179 \\
\end{align}\]
Hence, the 12th term from the end of this arithmetic progression is 179.

NOTE: -
Another method of solving this question is as follows
The 12th term of the sequence is also the 89th term of the sequence from the start.
Hence, we can find the answer by this method also.