Find the next five terms of the following sequences given by:

a_{1} = 4, a_{n} = 4a_{n−1} + 3, n > 1.

#### Solution

`a_1 = 4, a_n = 4a_(n - 1) + 3, n > 1`

`Here, we are given that *n* > 1`

So the next five terms of this A.P would be `a_2, a_3, a_4, a_5, a_6`

Now `a_1 = 4` .....(1)

So, to find the `a_2` term we use n = 2 we get

`a_2 = 4a_(2 - 1) + 3`

`a_2 = 4a_1 + 3`

`a_2 = 4(4) + 3` (Using 1)

`a_2 = 19` ....(2)

For `a_3` Using n = 3 we get

`a_3 = 4a_(3 -1) + 3` (Using 2)

`a_3 = 4a_2 + 3`

`a_3 = 4(19) + 3`

`a_3 = 79` .....(3)

For `a_4` using n = 4 we get

`a_4 = 4a_(4 -1) + 3`

`a_4 = 4a_3 + 3`

`a_4 = 4(79) + 3` (Using 4)

`a_4 = 319` ....(4)

For `a_5` using n = 5 we get

`a_5 = 4a_(5 - 1) + 3`

`a_5 = 4a_4 + 3`

`a_5 = 1279` ....(5)

For `a_6` using n = 6 we get

`a_6 = 4a_(6 -1) + 3`

`a_6 = 4a_5 + 3`

`a_6 = 4(1279) + 3` (Using 5)

`a_6 = 5119`

Therefore, the next five terms, of the given A.P are

`a_2= 19 , a_3 = 79, a_4 = 319, a_5 = 1279, a_6 = 5119`