Question

Given the explicit expression f (n) = 5n - 2, write a set showing the sequence, and write a recursive expression.

Answer 1

Answer:

$f(n) = 5 + f(n - 1)$

Step-by-step explanation:

Given

$f(n) = 5n - 2$

Required

Write a recursive expression

$f(n) = 5n - 2$

When n = 1

$f(1) = 5 * 1 - 2 = 5 - 2 = 3$

$f(1) = 3$

When n = 2

$f(2) = 5 * 2 - 2 = 10 - 2 = 8$

$f(2) = 8$ can be rewritten as

$f(2) = 5 + 3$

Substitute 3 for f(1)

$f(2) = 5 + f(1)$

Express 1 as 2 - 1

$f(2) = 5 + f(2-1)$ ----- (1)

When n = 3

$f(3) = 5 * 3 - 2 = 15 - 2 = 13$

$f(3) = 13$ can be rewritten as

$f(3) = 5 + 8$

Substitute 8 for f(2)

$f(3) = 5 + f(2)$

Express 2 as 3 - 1

$f(3) = 5 + f(3 - 1)$  ------ (2)

Write out (1) and (2)

$f(2) = 5 + f(2-1)$

$f(3) = 5 + f(3 - 1)$

Replace 2 and 3 with n in both cases; This gives

$f(n) = 5 + f(n - 1)$

Hence;

The recursive is $f(n) = 5 + f(n - 1)$