Answer :
Recursive formula of the sequence → [tex]T_n=10240(\frac{1}{4})^{n-1}[/tex]
Explicit formula of the sequence → [tex]T_1=10240, \\T_{n+1}=T_n(0.25)[/tex]
Height of the ball after 8th bounce will be 0.625 cm.
Geometric sequence and its explicit and recursive formula,
- If a sequence is in the form of [tex]ar,ar,ar^2,ar^3.....[/tex] n terms,
Sequence will be the geometric sequence and the explicit formula
of the sequence will be,
[tex]T_n=ar^{n-1}[/tex]
Here, [tex]a=[/tex] First term
[tex]r=[/tex] Common ratio
[tex]n=[/tex] Number of terms
- Similarly, recursive formula will be,
[tex]a=[/tex] First term
[tex]T_{n+1}=T_n(r)[/tex]
From the table attached,
There is common ratio between the successive term and previous term under the column (height after the bounce) is,
[tex]\frac{T_2}{T_1}=\frac{2560}{10240}=\frac{1}{4}[/tex]
[tex]\frac{T_3}{T_2}=\frac{640}{2560}=\frac{1}{4}[/tex]
Therefore, height of the ball after each bounce will represent a geometric sequence.
Explicit formula of a geometric sequence will be given by the expression,
[tex]T_n=10240(\frac{1}{4})^{n-1}[/tex]
Recursive formula for the sequence will be,
[tex]T_1=10240[/tex]
[tex]T_{n+1}=T_n(\frac{1}{4})[/tex]
By using recursive formula,
Height of the ball after 8th bounce,
[tex]T_8=10240(\frac{1}{4})^{8-1}[/tex]
[tex]=10240(0.25)7[/tex]
[tex]=0.625[/tex] cm
Therefore, explicit formula to represent the height of the ball will be [tex]T_n=10240(\frac{1}{4})^{n-1}[/tex]recursive formula will be, [tex]T_1=10240\\T_{n+1}=T_n(0.25)[/tex]
And height of the ball after 8th bounce will be 0.625 cm.
Learn more about the geometric sequence here,
https://brainly.com/question/25250161?referrer=searchResults