Using the index of a sequence as the domain and the value of the sequence as the range, is a sequence a functi

Question

Using the index of a sequence as the domain and the value of the sequence as the range, is a sequence a function?

A) Which one of the basic functions (linear, quadratic, rational, or exponential) is related to the arithmetic sequence?

B)Which one of the basic functions (linear, quadratic, rational, or
exponential) is related to the geometric sequence?

C)Give at least two real-life examples of a sequences or series. One example should be arithmetic, and the second should be geometric. Explain how these examples would affect you personally.

Answer 1

Yes, a sequence is a function. The reason is that, for each "input" (index), there is at most one "output" (value).

So for instance, the arithmetic sequence:

1, 3, 5, 7, 9, ....

Can be written in function form as:

a(n) = 1 + 2n

If your indices start at 0, or:

a(n) = -1 + 2n

If your indices start at 1.

A) An arithmetic sequence looks like:

a(n) = a(0) + kn

This is the same form as a linear equation:

y(x) = b + mx

So arithmetic sequences are related to linear functions.

B) A geometric sequence is of the form:

a(n) = a(0)rⁿ

This is the same form as an exponential equation:

y(x) = kb^x

So geometric sequences are related to exponential functions.

C) A real-world example of an arithmetic sequence might be buying eggs at the store. Eggs come in dozens, so if you buy 1 package, you get 12 eggs. If you buy 2 packages, you get 24 eggs. If you buy 3 packages, you get 36 eggs... This is an arithmetic sequence where k = 12.

A real-world example of a geometric sequence is compound interest. If you invest $100 in an account that pays 5% per year, after the first year you would have $105. After the second year, you would have $110.25. After the third year, you'd have $115.75. This is a geometric sequence where r = 1.05.