# 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 $$n \ge1$$.

1. Explain why $$f(1) = \text- 20 - 5$$.
2. Explain why $$f(3) = \text- 20 - 5 - 5 - 5$$.
3. 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 $$n\ge2$$.

1. Find the first 5 terms of the sequence.
2. Graph the value of the term as a function of the term number.
3. Is the sequence arithmetic, geometric, or neither? Explain how you know.
(From Unit 1, Lesson 7.)

### Problem 4

Here is a graph of sequence $$M$$. Define $$M$$ recursively using function notation.

(From Unit 1, Lesson 6.)

### Problem 5

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

1. $$a(1) = 5, a(n) = a(n-1) + 3$$ for $$n\ge2$$.
2. $$b(1) = 1, b(n) = 3 \boldcdot b(n-1)$$ for $$n\ge2$$.
3. $$c(1) = 3, c(n) = \text-c(n-1) + 1$$ for $$n\ge2$$.
4. $$d(1) = 5, d(n) = d(n-1) + n$$ for $$n\ge2$$.
(From Unit 1, Lesson 5.)

### Problem 6

Here is the graph of a sequence:

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