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.
Blank coordinate plane with grid, origin O. Horizontal axis from 0 to 6 by 1’s, labeled “term number”. Vertical axis from 0 to 60 by 5’s, labeled “value”.
(From Unit 1, Lesson 7.)

Problem 4

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

Graph of a sequence M.
(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.
Graph of sequence on coordinate plane.
(From Unit 1, Lesson 7.)