Question

The sequence 6, 18, 54, 162, ... shows the number of pushups Kendall did each week, starting with her first week of exercising.

(a) Is this an arithmetic or geometric sequence?
(b) How can you find the next number in the sequence?
(c) Give the rule you would use to find the 20th week.
(A rule is like an equation, e.g. = 2 + 90)

Answer 1

(a) geometric

(b) multiply the previous value by 3

(c) p(n) = 2×3^n

Explanation:

(a) The numbers don't have a common difference, but they have a common ratio:

18/6 = 3 = 54/18 = 162/54

so the sequence is geometric.

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(b) As we found in part (a), each term is 3 times the previous one. To find the next term, multiply the current term by 3.

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(c) The general rule for the n-th term of a geometric sequence is ...

an = a1·r^(n-1)

where a1 is the first term and r is the common ratio. Here, we have a1=6 and r=3, so pushups for week n can be described by ...

p(n) = 6×3^(n-1)

p(n) = 2×3^n . . . . . . simplifying

This is the formula that can be used to find the 20th term of the sequence.

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Note that p(20) = 6,973,568,802. If Kendall does that many pushups in a week, she is doing more than 11,000 per second without taking any break for the week.