Find the next five terms of each of the following sequences given by :
$ a_{1}=4, a_{n}=4 a_{n-1}+3, n>1 $

Given:

\( a_{1}=4, a_{n}=4 a_{n-1}+3, n>1 \)
To do:

We have to find the next five terms of the given sequence.
Solution: 

The next five terms in the sequence are obtained by substituting $n=2, 3, 4, 5, 6$ respectively.

When $n=2$,

$a_2=4a_{2-1}+3$

$=4a_1+3$

$=4(4)+3$

$=16+3$

$=19$

When $n=3$,

$a_3=4a_{3-1}+3$

$=4a_2+3$

$=4(19)+3$

$=76+3$

$=79$

When $n=4$,

$a_4=4a_{4-1}+3$

$=4a_3+3$

$=4(79)+3$

$=316+3$

$=319$

When $n=5$,

$a_5=4a_{5-1}+3$

$=4a_4+3$

$=4(319)+3$

$=1276+3$

$=1279$

When $n=6$,

$a_6=4a_{6-1}+3$

$=4a_5+3$

$=4(1279)+3$

$=5116+3$

$=5119$

Therefore, the next five terms of the given sequence are $19, 79, 319, 1279$ and $5119$. 

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Updated on: 10-Oct-2022

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