3. The sequence 6, 18, 54, 162, … shows the number of pushups Kendall did each week, starting with her first week of exercising. (a) What is the recursive rule for the sequence? (b) What is the iterative rule for the sequence?

a) The recursive rule for the sequence is

b) The iterative rule for the sequence is

The given sequence is:

6, 18, 54, 162, …

The common ratio, r = 18/6

r  =  3

Let the last term be

Let the preceding term be

The recursive rule for the sequence is given as:

b) The iterative rule is gotten by using the formula:

Substitute a = 6 and r = 3 into the formula above

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Please check the explanation.

Step-by-step explanation:

(a) What is the recursive rule for the sequence?

Given the sequence

6, 18, 54, 162,...

Please note that each successive term is 3 times the previous term.

i.e. 18 = 6×3, 54 = 18×3, 162 = 54×3

Here, r = 3 is the common multiplier

In other words, every next term can be obtained by multiplying the previous term by a number 'r', using the recursive formula

Here, aₙ is r times the previous term aₙ₋1.

For example, in order to determine the 2nd term a₂, just put n=2 in the recursive formula

so substituting r = 3, and n = 1 in the recursive formula

as

a₁ = 6

so

(b) What is the iterative rule for the sequence?

Given the sequence

6, 18, 54, 162,...

A geometric sequence has a constant ratio 'r' and is defined as

The ratio of all the adjacent terms is the same and equal to, so

r = 3

as

a₁ = 6

so substituting r = 3 and a₁ = 6 in the nth term of the sequence

Thus, the iterative rule for the sequence is

(f - g)(x) = 5x^3 - 3

step-by-step explanation:

rearrange these functions vertically, with like terms in separate columns:

f(x) = 5x^3 - 2

g(x) =         +1

now subtract, column by column:

(f - g)(x) = 5x^3 - 3