Write the first five terms of each of the sequence and obtain the corresponding series :

$a_{1} = 3, a_{n} = 3 a_{n - 1} + 2 for all n > 1$

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Solution

Here $a_{1} = 3, and a_{n} = 3 a_{n - 1} + 2$

Putting n = 2, 3, 4 and 5, we have

$a_{2} = 3 a_{2 - 1} + 2 = 3 a_{1} + 2 = 3 \times 3 + 2$ $[∴ a_{1} = 3]$

$= 9 + 2 = 11$

$a_{3} = 3 a_{3 - 1} + 2 = 3 a_{2} + 2 = 3 \times 11 + 2 = 33 + 2 = 35 [∴ a_{2} = 11]$

$a_{4} = 3 a_{4 - 1} + 2 = 3 a_{3} + 2 = 3 \times 35 + 2 = 105 + 2 = 107 [∴ a_{3} = 35]$

$a_{4} = 3 a_{5 - 1} + 2 = 3 a_{4} + 2 = 3 \times 107 + 2 = 321 + 2 = 323 [∴ a_{4} = 107]$

Thus, first five terms of the sequences are 3, 11, 35, 107 and 232.

The corresponding series is

3 + 11 + 35 + 107 + 323 + .......

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a_{1} = 3, a_{n} = 3 a_{n - 1} + 2

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