Question

Jia is skipping a rock across a pond. The sequence {4.2, 3.57, 3.0345, 2.5793, …} describes the height of the rock on each successive skip. Which explicit formula describes this geometric sequence?

Answer 1

Answer:

`{ \boxed{\tt {a _(n) = 4.94* 0.85{}^(n) }}}`

Explanation:

» General explicit formula for geometric sequence:

`{ \tt{a _(n) = ar {}^(n - 1) }} \\`

• a → first term
• a_n → nth term
• r → common ratio

» In the sequence given;

• n → 4
• a_1 → 4.2
• r → 4.2/3.57 → 17/20

`{ \tt{a _(n) = 4.2 * {( (17)/(20)) }^(n- 1) }} \\ \\ { \tt{{a _(n) = 4.2 * ( (17)/(20)) {}^(n) * ( (17)/(20)) {}^( - 1) }}} \\ \\ { \tt{a _(n) = 4.2 * ( (17)/(20)) {}^(n) * (20)/(17) }} \\ \\{ \tt {a _(n) = 4.94* 17/20 {}^(n) }}`

Answer 2

`u_n=4.2(0.85)^(n-1)`

Explanation:

`u_1=4.2\\u_2=3.57\\u_3=3.0345\\u_4=2.5793`

Geometric formula sequence:
`u_n=ar^((n-1))`

(where
`a` is the first term of the sequence and
`r` is the common ratio)

To find the common ratio, divide one of the terms by the previous term:

`r=(u_2)/(u_1) =(3.57)/(4.2) =0.85`

From inspection,
`a=4.2`

Therefore,
`u_n=4.2(0.85)^(n-1)`