Question

15. Convert the following recursive formula to an explicit formula.a = 57: -3 . an-1

Answer 1

The recursive formula is

a = 57 ------- first term and d = -3 ------- common difference

$a_n=-3+a_{n-1}$

The explicit formula of the sequence is

$a_n=a+(n-1)d$

Where d is the constant difference between each 2 consecutive terms

∵ a is the first term

∴ a = 57

∵ d is the constant difference

∴ d = -3

→ Substitute them in the form above

$\therefore a_n=57+(n-1)\times(-3)$

Let us simplify it

$\begin{gathered} \because a_n=57+n(-3)-1(-3) \\ \therefore a_n=57-3n+3 \\ \therefore a_n=(57+3)-3n \\ \therefore a_n=60-3n \end{gathered}$

Then the explicit formula is an = 60 - 3n