Question

. A sequence starts with 180, 120, …. If the sequence is arithmetic, create a recursive and explicit formula for the sequence. If the sequence is geometric, create a recursive and explicit formula for the sequence.

Answer 1

If the sequence is arithmetic, the recursive formula is a(n) = a(n-1) - d and the explicit formula is a(n) = a(1) + (n-1)d. The given sequence is not geometric.

Step-by-step explanation:

If the sequence is arithmetic, the recursive formula can be written as:

a(n) = a(n-1) - d

where a(n) is the nth term in the sequence, a(n-1) is the previous term in the sequence, and d is the common difference between the terms.

The explicit formula can be written as:

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

where a(1) is the first term in the sequence and n is the position of the term in the sequence.

Since the given sequence starts with 180, 120, it is not a geometric sequence.

Answer 2

Arithmetic Sequence

`T_n = 240-60n` ---- Explicit

`T_n = T_(n-1) - 60` --- Recursive

Geometric Sequence

`T_n = 270* ((2)/(3))^n` ---- Explicit

`T_n = T_(n-1) * (2)/(3)` ---- Recursive

Step-by-step explanation:

Given

`180, 120,....`

(a) Assume it is an arithmetic sequence

The explicit formula is calculated using:

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

Where

`a = 180`

`d = 120 - 180`

`d = -60`

So, we have:

`T_n = 180 + (n - 1)*-60`

`T_n = 180 -60n + 60`

Rewrite

`T_n = 180 + 60-60n`

`T_n = 240-60n`

The recursive function is calculated using:

`T_1 = 180`

`T_2 = 120 = 180 - 60 = T_1 - 60`

`T_3 = 60 = 120 - 60 = T_2 -60`

-

`T_n = T_(n-1) - 60`

(b) Assume it is a geometric sequence

The explicit formula is calculated using:

`T_n = ar^(n-1)`

Where

`a = 180`

`r = (120)/(180)`

`r = (2)/(3)`

So, we have:

`T_n = 180 * ((2)/(3))^(n-1)`

Split

`T_n = 180 * ((2)/(3))^n / ((2)/(3))^1`

`T_n = 180 * ((2)/(3))^n / ((2)/(3))`

Rewrite as:

`T_n = 180 * ((2)/(3))^n * ((3)/(2))`

`T_n = 180 * ((3)/(2))* ((2)/(3))^n`

`T_n = 180 * (3)/(2)* ((2)/(3))^n`

`T_n = 90 * 3* ((2)/(3))^n`

`T_n = 270* ((2)/(3))^n`

The recursive function is calculated using:

`T_1 = 180`

`T_2 = 120 = 180 * (2)/(3) = T_1 * (2)/(3)`

`T_3 = 80 = 120 * (2)/(3) = T_2 * (2)/(3)`

-

`T_n = T_(n-1) * (2)/(3)`