#### Question

Find the 12^{th} term from the end of the following arithmetic progressions:

3, 5, 7, 9, ... 201

#### Solution

In the given problem, we need to find the 12^{th} term from the end for the given A.P.

3, 5, 7, 9, ... 201

Here, to find the 12^{th} term from the end let us first find the total number of terms. Let us take the total number of terms as *n*.

So

First term (*a*) = 3

Last term (a_{n}) = 201

Common difference (d) = 5 - 3

=2

Now as we know

`a_n = a + (n - 1)d`

So for the last term

201 = 3 + (n - 1)2

201 = 3 + 2n - 2

201 = 1 + 2n

201 -1 = 2n

Furthur simplifying

200 = 2n

`n = 200/2`

n = 100

So, the 12^{th} term from the end means the 89^{th} term from the beginning.

So, for the 89^{th} term (*n =* 89)

`a_89 = 3 + (89 - 1)2`

= 3 + (88)2

= 3 + 176

= 179

Therefore the 12th term from the end of the given A.P is 179