Find the next five terms of each of the following sequences given by:
$a_{1} = a_{2} = 2, a_{n} = a_{n - 1} - 3, n > 2$

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Solution

Given,
The $n^{t h}$ term of the sequence is
$a_{n} = a_{n - 1} - 3 f o r n > 2 . . . . . . . . . . . . . e q (1)$
And
$a_{1} = a_{2} = 2 . . . . . e q (2)$

1) For finding the third term of sequence $(a_{3})$ put n=3 in eq(1)
$⟹ a_{3} = a_{3 - 1} - 3$
$⟹ a_{3} = a_{2} - 3$
put value of $a_{2}$ from eq(2)
$⟹ a_{3} = 2 - 3$
$⟹ a_{3} = - 1 . . . . . . . e q (3)$

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

3) For finding the fifth term of sequence $(a_{5})$ put n=5 in eq(1)
$⟹ a_{5} = a_{5 - 1} - 3$
$⟹ a_{5} = a_{4} - 3$
put value of $a_{4}$ from eq(4)
$⟹ a_{5} = - 4 - 3$
$⟹ a_{5} = - 7 . . . . e q (5)$

4) For finding the sixth term of sequence $(a_{6})$ put n=6 in eq(1)
$⟹ a_{6} = a_{6 - 1} - 3$
$⟹ a_{6} = a_{5} - 3$
put value of $a_{5}$ from eq(5)
$⟹ a_{6} = - 7 - 3$
$⟹ a_{6} = - 10 . . . . e q (6)$

5)For finding the seventh term of sequence $(a_{7})$ put n=7 in eq(1)
$⟹ a_{7} = a_{7 - 1} - 3$
$⟹ a_{7} = a_{6} - 3$
put value of $a_{6}$ from eq(6)
$⟹ a_{7} = - 10 - 3$
$⟹ a_{7} = - 13 . . . . . . e q (7)$

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