# Lesson 8

The $n^{\text{th}}$ Term

- Let’s see how to find terms of sequences directly.

### Problem 1

A sequence is defined by \(f(0) = \text-20, f(n) = f(n-1) -5\) for \(n \ge1\).

- Explain why \(f(1) = \text- 20 - 5\).
- Explain why \(f(3) = \text- 20 - 5 - 5 - 5\).
- Complete the expression: \(f(10)=\text-20-\underline{\hspace{.5in}}\). Explain your reasoning.

### Problem 2

A sequence is defined by \(f(0) = \text- 4, f(n) = f(n-1) - 2\) for \(n\ge1\). Write a definition for the \(n^{\text{th}}\) term of the sequence.

### Problem 3

Here is the recursive definition of a sequence: \(f(1) = 3,f(n) = 2 \boldcdot f(n-1)\) for \(n\ge2\).

- Find the first 5 terms of the sequence.
- Graph the value of the term as a function of the term number.
- Is the sequence arithmetic, geometric, or neither? Explain how you know.

### Problem 4

Here is a graph of sequence \(M\). Define \(M\) recursively using function notation.

### Problem 5

Write the first five terms of each sequence. Determine whether each sequence is arithmetic, geometric, or neither.

- \(a(1) = 5, a(n) = a(n-1) + 3\) for \(n\ge2\).
- \(b(1) = 1, b(n) = 3 \boldcdot b(n-1)\) for \(n\ge2\).
- \(c(1) = 3, c(n) = \text-c(n-1) + 1\) for \(n\ge2\).
- \(d(1) = 5, d(n) = d(n-1) + n\) for \(n\ge2\).

### Problem 6

Here is the graph of a sequence:

- Is this sequence arithmetic or geometric? Explain how you know.
- List at least the first five terms of the sequence.
- Write a recursive definition of the sequence.