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

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Solution

Given,
The nth term of the sequence is

$a_{n} = 4 a_{n - 1} + 3 f o r n > 1 . . . . . . . . . . . . . e q (1)$
And
$⟹ a_{1} = 4 . . . . . . . e q (2)$

1) For finding the second term of sequence $(a_{2})$ put n=2 in eq(1)
$⟹ a_{2} = 4 a_{2 - 1} + 3$
$⟹ a_{2} = 4 \times a_{1} + 3$
put value of $a_{1}$ from eq(2)
$⟹ a_{2} = 4 \times 4 + 3$
$⟹ a_{2} = 16 + 3$
$⟹ a_{2} = 19 . . . . e q (3)$

2) For finding the third term of seqence $(a_{3})$ put n=3 in eq(1)
$⟹ a_{3} = 4 a_{3 - 1} + 3$
$⟹ a_{3} = 4 \times a_{2} + 3$
put value of $a_{2}$ from eq(3)
$⟹ a_{3} = 4 \times 19 + 3$
$⟹ a_{3} = 76 + 3$
$⟹ a_{3} = 79 . . . . e q (4)$

3)For finding the fourth term of sequence $(a_{4})$ put n=4 in eq(1)
$⟹ a_{4} = 4 a_{4 - 1} + 3$
$⟹ a_{4} = 4 \times a_{3} + 3$
put value of $a_{3}$ from eq(4)
$⟹ a_{4} = 4 \times 79 + 3$
$⟹ a_{4} = 316 + 3$
$⟹ a_{4} = 319 . . . . e q (5)$

4) For finding the fifth term of sequence $(a_{5})$ put n=5 in eq(1)
$⟹ a_{5} = 4 a_{5 - 1} + 3$
$⟹ a_{5} = 4 \times a_{4} + 3$
put value of $a_{4}$ from eq(5)
$⟹ a_{5} = 4 \times 319 + 3$
$⟹ a_{5} = 1276 + 3$
$⟹ a_{5} = 1279 . . . . e q (6)$

5) For finding the sixth term of sequence $(a_{6})$ put n=6 in eq(1)
$⟹ a_{6} = 4 a_{6 - 1} + 3$
$⟹ a_{6} = 4 \times a_{5} + 3$
put value of $a_{5}$ from eq(6)
$⟹ a_{6} = 4 \times 1279 + 3$
$⟹ a_{6} = 5116 + 3$
$⟹ a_{6} = 5119 . . . . e q (7)$

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