Question

f(3) = 12 for a geometric sequence that is defined

recursively by the formula f(n) = 0.5 ∙ f(n – 1), where

n is an integer and n > 0.

Write the equation in explicit form.

Answer 1

Answer:

$f(n) = 96 * 0.5^n$

Step-by-step explanation:

Given

$f(3) = 12$

$f(n) = 0.5 * f(n-1)$

Required

Write as an explicit function

When n = 3, we have:

$f(3) = 12$

$f(3) = 0.5 * f(3-1)$

$f(3) = 0.5 * f(2)$

Substitute: $f(3) = 12$

$12 = 0.5 * f(2)$

Solve for f(2)

$f(2) = 12/0.5$

$f(2) = 24$

So, we have;

$f(3) = 12$

$f(2) = 24$

Since it is a geometric sequence, calculate the common ratio (r)

$r = \frac{f(3)}{f(2)}$

$r = \frac{12}{24}$

$r = 0.5$

Calculate the first term using:

$r = \frac{f(2)}{f(1)}$

Solve for f(1)

$f(1) = \frac{f(2)}{r}$

$f(1) = \frac{24}{0.5}$

$f(1) = 48$

The explicit function is then calculated as:

$f(n) = f(1) * r^{n-1$ ----nth term of a gp

$f(n) = 48 * 0.5^{n-1$

Split the exponent

$f(n) = 48 * \frac{0.5^n}{0.5^1}$

$f(n) = 48 * \frac{0.5^n}{0.5}$

$f(n) = 96 * 0.5^n$