How do you find the fourth term in the following recursively defined sequence given $a_{1} = 4$ and $a_{n} = a_{n} - 1 + 2$ ?

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Solution

Let the sequence $a_{n}$ be defined as follows:
$a_{1} = 4 and a_{n} = a_{n} - 1 + 2$
It is given that
$a_{1} = 4$
We use this formula
$a_{n =} a_{n - 1} + 2$
Put $n = 2 in (1)$
$a_{2} = a_{1} + n$
$a_{2} = 4 + 2 = 6$
Now,
$a_{3} = a_{2} + n$
$a_{3} = 6 + 2 = 8$
Similarly, we get
$a_{4} = a_{3} + n$
$a_{4} = 8 + 2 = 10$
Therefore ,the fourth term is $a_{4} = 10$

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