Study Guide for Final Exam

This will be an open-book final exam. A graphing calculator is permitted for this exam.

You will have 2 hours to complete the exam. The exam will start promptly at 11:35 AM, so please start on time.

There will be 7 problems on the exam: 4 modeled after problems from the previous exams, and 3 from the material covered since Exam 4.

To do well on the new material on the exam, you should be able to do the following:

Section 12.1: Sequences and Summation Notation

  1. Recognize the notation \(a_{n} := a(n)\) for a sequence as a function whose domain is the natural numbers \(n \in \mathbb{N}\), i.e. \(n\) is one of the ‘counting numbers’ \(1, 2, 3, \ldots\).
  2. Given a closed-form expression for a sequence \(a_{1}, a_{2}, \ldots, a_{n}, \ldots\), find the first \(N\) terms of the sequence.
  3. Given several terms of a sequence that can be expressed in a closed-form, determine the closed-form expression for the sequence.
  4. Explain what it means for a sequence to be recursively defined.
  5. Given a recursively defined sequence, determine the first \(N\) terms of the sequence.
  6. Find the \(n\text{th}\) partial sum \(S_{n}\) of a sequence.
  7. Recognize the \(n\text{th}\) partial sum \(S_{1}, S_{2},\ldots, S_{n}, \ldots\) of a sequence as itself a sequence.
  8. Recognize and use summation notation, aka Sigma notation, aka \(S_{n} = \displaystyle\sum_{k = 1}^{n} a_{k}\), for the \(n\text{th}\) partial sum \(S_{n}\).
  9. Use sequences and partial sums to model situations that require discrete changes to some quantity.

Section 12.2: Arithmetic Sequences

  1. Specify what makes an arithmetic sequence ‘arithmetic.’
  2. Identify from three or more terms of a sequence, whether the sequence is arithmetic, and if the sequence is arithmetic, identify the first term \(a\) and common difference \(d\).
  3. Identify and use the ‘trick’ for determining the \(n\)-th partial sum of an arithmetic sequence by appropriately pairing up the terms in the arithmetic sequence.
  4. Given a situation that can be modeled using an arithmetic sequence or the partial sum of an arithmetic sequence, identify the appropriate arithmetic sequence and / or its partial sum.

Section 12.3: Geometric Sequences

  1. Specify what makes a geometric sequence ‘geometric.’
  2. Identify from three or more terms of a sequence, whether the sequence is geometric, and if the sequence is geometric, identify the first term \(a\) and common ratio \(r\).
  3. Identify and use the ‘trick’ for determining the \(n\)-th partial sum of a geometric sequence by appropriately pairing up the terms of the geometric sequence and the terms of the geometric sequence times \(r\).
  4. Given a situation that can be modeled using a geometric sequence or the partial sum of geometric sequence, identify the appropriate geometric sequence and / or its partial sum.