Question

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

(i) 3, 5, 7, 9, ... 201

(ii) 3, 8, 13, ..., 253

(iii) 1, 4, 7, 10, ..., 88

Answer 1

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

(i) 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) =

Now, as we know,

So, for the last term,

Further simplifying,

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

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

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

(ii) 3, 8, 13 …253

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) = 253

Common difference, d =

Now, as we know,

So, for the last term,

Further simplifying,

So, the 12^th term from the end means the 40^th term from the beginning.

So, for the 40^th term (n = 40)

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

(iii) 1, 4, 7, 10 …88

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) = 1

Last term (a_n) = 88

Common difference, d= =3

Now, as we know,

So, for the last term,

Further simplifying,

So, the 12^th term from the end means the 19^th term from the beginning.

So, for the 19^th term (n = 19)

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