The graph represents the first 10 terms of a sequence....

The graph represents the first 10 terms of a sequence. Complete each expression for the apparent $n$ th term $\left(a_{n}\right)$ of the sequence. Which expressions are appropriate to represent the $\operatorname{cost} a_{n}$ to buy $n$ MP3 songs at a cost of $\$ 1$ per song? Explain. (a) $a_{n}=1$ (b) $a_{n}=\frac{1}{(n-1) !}$ (c) $a_{n}=\sum_{k=1}^{n}$

The graph represents the first 10 terms of a sequence. Complete each expression for the apparent $n$ th term $\left(a_{n}\right)$ of the sequence. Which expressions are appropriate to represent the $\operatorname{cost} a_{n}$ to buy $n$ MP3 songs at a cost of $\$ 1$ per song? Explain.
(a) $a_{n}=1$
(b) $a_{n}=\frac{1}{(n-1) !}$
(c) $a_{n}=\sum_{k=1}^{n}$

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Key Concepts

Sequence

A sequence is an ordered list of numbers defined by a particular rule or formula. Understanding how sequences work helps in modeling a series of values like costs that depend on the position in the sequence (in this case, the number of songs purchased).

Constant Function

A constant function yields the same output regardless of the input. Although the cost of one song is constant at $1, the total cost for n songs should be represented as n times the constant cost, not merely the constant value.

Summation Notation

Summation notation is a compact way to denote the addition of a sequence of terms. When computing a total cost by adding a fixed cost for each song over n songs, summation notation (such as ? from k = 1 to n of 1) clearly communicates the process of adding $1 repeatedly, which results in a linear function of n.

Factorial Function

The factorial function, which multiplies a series of descending natural numbers, usually introduces non?linear growth. It is generally used in contexts involving permutations and combinations, and does not appropriately model a situation where each additional unit (song) adds a constant cost.

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