Question

A sequence is defined by this recursive function:

f(n) = (2.5) · f(n – 1); f(1) = 48

Complete the table.
Index Function Notation Term Value
1 f(1) = 48 48
2 f(2) = (2.5) · f(1) 120
3 f(3) = (2.5) · f(2) 300
4 f(4) = (2.5) · f(3) ___
5 f(5) = (2.5) · f(4) ___
6 f(6) = (2.5) · f(5) ___

To the nearest hundredth, the tenth term in the sequence is ___

Answer 1

750

1875

29,296.88

Explanation:

Use the values from the table, and find the terms one after the other:

f(4) = 2.5 • f(3) = 2.5(300) = 750

f(5) = 2.5 • f(4) = 2.5(750) = 1,875

To find the eighth term, we must first find the sixth and seventh terms. So, use the fifth term to find the sixth term.

f(6) = 2.5 • f(5) = 2.5(1,875) = 4,687.5

Then, find the seventh term.

f(7) = 2.5 • f(6) = 2.5(4,687.5) = 11,718.75

Finally, find the eighth term.

f(8) = 2.5 • f(7) = 2.5(11,718.75) = 29,296.875

Rounded to the nearest hundredth, the eighth term of the sequence is 29,296.88.

Answer 2

Hello from MrBillDoesMath!

Answer:

See the Discussion section below.

Discussion:

Study this pattern:

f(1) = 48

f(2) = 2.5 * 48

f(3) = 2.5 f(2) = 2.5 * ( 2.5 * 48 ) = 2.5^2 * 48

f(4) = 2.5 * f(3) = 2.5 * (2.5^2 * 48) = 2.5^3 * 48

so generally.....

f(n) = 2.5^(n-1) * 48, n >=1

Therefore

f(4) = 2.5^3 * 48 = 750

f(5) = 2.5^4 * 48 = 1875

f(6) = 2.5^5 * 48 = 4687.5

f(10) = 2.5^9 * 48 = 183105.46875 which is 183105.47 to the nearest hundredth

Thank you,

MrB