Find the first five terms of the sequence given by
$a_{1} = 2, a_{2} = 3 + a_{1}$ and $a_{n} = 2 a_{n - 1} + 5$ for $n > 2$

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Solution

The given sequence is $a_{n} = 2 a_{n - 1} + 5$ where the first term $a_{1} = 2$ and the second term is $a_{2} = 3 + a_{1}$ .

Substitute $a_{1} = 2$ in $a_{2} = 3 + a_{1}$ as shown below:

$a_{2} = 3 + 2 = 5$

Therefore, the second term of the given sequence is $a_{2} = 5$

Now, the next three terms of the sequence $a_{n} = 2 a_{n - 1} + 5$ can be determined as follows:

$a_{3} = 2 a_{3 - 1} + 5 = 2 a_{2} + 5 = (2 \times 5) + 5 = 10 + 5 = 15$

$a_{4} = 2 a_{4 - 1} + 5 = 2 a_{3} + 5 = (2 \times 15) + 5 = 30 + 5 = 35$

$a_{5} = 2 a_{5 - 1} + 5 = 2 a_{4} + 5 = (2 \times 35) + 5 = 70 + 5 = 75$

Hence, the first five terms of the sequence $a_{n} = 2 a_{n - 1} + 5$ are $2, 5, 15, 35, 75, . . . .$

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