Question

Negative 2 and two-thirds, negative 5 and one-third, negative 10 and two-thirds, negative 21 and one-third, negative 42 and two-thirds, ellipsis Which formula can be used to describe the sequence?

Answer 1

Answer:

$f(n) = -\frac{8}{3}n$

Step-by-step explanation:

Given

$-2\frac{2}{3}, -5\frac{1}{3}, -10\frac{2}{3}, -21\frac{1}{3}, -42\frac{2}{3}$

Required

The explicit formula

The above sequence is an arithmetic sequence and it is bounded by:

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

Where

$a = -2\frac{2}{3}$ -- the first term

$d = f(2) -f(1)$

So, we have:

$d = -5\frac{1}{3} - -2\frac{2}{3}$

$d = -5\frac{1}{3} +2\frac{2}{3}$

Express as improper fraction

$d = -\frac{16}{3} +\frac{8}{3}$

Take LCM

$d = \frac{-16+8}{3}$

$d = -\frac{8}{3}$

So, we have:

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

$f(n) = -2\frac{2}{3} + (n - 1) * -\frac{8}{3}$

Express all fractions as improper

$f(n) = -\frac{8}{3} + (n - 1) * -\frac{8}{3}$

Open brackets

$f(n) = -\frac{8}{3} -\frac{8}{3}n +\frac{8}{3}$

Collect like terms

$f(n) = -\frac{8}{3}n +\frac{8}{3}-\frac{8}{3}$

$f(n) = -\frac{8}{3}n$