Question

Use the first five terms of sequence H to define the sequence recursively using function notation.

2.5, 7.5, 22.5, 67.5, 202.5, . . .

Answer 1

Explanation:

Let H be the sequence given by:

H = {2.5, 7.5, 22.5, 67.5, 202.5, ...}

To define the sequence recursively using function notation, we can use the formula for the nth term of the sequence, based on the previous term:

H(1) = 2.5 (given)

H(n) = 3H(n-1) + 2, for n > 1

This means that to find any term in the sequence after the first term, we multiply the previous term by 3 and add 2.

Using the recursive formula, we can find the next few terms of the sequence:

H(2) = 3H(1) + 2 = 3(2.5) + 2 = 10

H(3) = 3H(2) + 2 = 3(10) + 2 = 32

H(4) = 3H(3) + 2 = 3(32) + 2 = 98

H(5) = 3H(4) + 2 = 3(98) + 2 = 296

So the sequence defined recursively using function notation is:

H(1) = 2.5

H(n) = 3H(n-1) + 2, for n > 1

where H(n) is the nth term of the sequence.