Write the first five terms and obtain the corresponding series, whose sequence is $a_{1} = 3, a_{n} = 3 a_{n - 1} + 2$ for all $n > 1$

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Solution

Finding sequence and corresponding series.
Given, $a_{1} = 3, a_{n} = 3 a_{n - 1} + 2$ for all $n > 1, n \in N$ .
Substituting $n = 2, 3, 4$ and $5$ in $a_{n}$ , we get

$a_{2} = 3 a_{2 - 1} + 2$
$\Rightarrow a_{2} = 3 a_{1} + 2$
$\Rightarrow a_{2} = 3 \times 3 + 2$
$\Rightarrow a_{2} = 11$

$a_{3} = 3 a_{3 - 1} + 2$
$\Rightarrow a_{3} = 3 a_{2} + 2$
$\Rightarrow a_{3} = 3 \times 11 + 2$
$\Rightarrow a_{3} = 35$

$a_{4} = 3 a_{4 - 1} + 2$
$\Rightarrow a_{4} = 3 a_{3} + 2$
$\Rightarrow a_{4} = 3 \times 35 + 2$
$\Rightarrow a_{4} = 107$

$a_{5} = 3 a_{5 - 1} + 2$
$\Rightarrow a_{5} = 3 a_{4} + 2$
$\Rightarrow a_{5} = 3 \times 107 + 2$
$\Rightarrow a_{5} = 323$

Hence, the first five terms of the sequence are $3, 11, 35, 107,$ and $323$ and corresponding series is $3 + 11 + 35 + 107 + 323 + \dots$ .

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Similar questions

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

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

a_{1} = - 1, a_{n} = \frac{a_{n - 1}}{n}, n \geq 2

a_{1} = a_{2} = 2, a_{n} = a_{n - 1} - 1, n > 2

a_{1} = a_{2} = 2, a_{n} = a_{n - 1} - 1, n > 2

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

n > 1

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