Question

​ g(1)=−19 g(n)=g(n−1)+6 ​ Find an explicit formula for g(n)g(n)g, left parenthesis, n, right parenthesis. g(n)=g(n)=g, left parenthesis, n, right parenthesis, equals

Answer 1

`g(n) = 6n -25`

Explanation:

Given:

`g(1) = -19`

`g(n) = g(n - 1) + 6`

Required

The explicit formula

Let n = 2; So, we have:

`g(n) = g(n - 1) + 6`

`g(2) = g(2 - 1) + 6`

`g(2) = g(1) + 6`

`g(2) = -19 + 6`

`g(2) = -13`

So, we have:

`g(1) = -19` ----- First term

`g(2) = -13`

Calculate common difference (d)

`d = g(2) - g(1)`

`d = -13 --19`

`d = 6`

The explicit function is then calculated as:

`g(n) = a + (n - 1)d`

Where

`a = -19` --- First term

So:

`g(n) = a + (n - 1)d`

`g(n) = -19 + (n - 1)*6`

Open bracket

`g(n) = -19 + 6n - 6`

Collect like terms

`g(n) = 6n - 6-19`

`g(n) = 6n -25`