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Calculus for Team-Based Inquiry Learning: 2024 Edition PREVIEW

TBIL Institute Fellows

Section 8.1 Sequence Formulas (SQ1)

Subsection 8.1.1 Activities

Activity 8.1.1.

Which of the following are sequences?
  1. monthly gas bill
  2. days in the year
  3. how long you wash dishes
  4. \(\displaystyle 1, 1, 2, 3, 5, 8, \ldots\)
  5. how much you spend on groceries

Activity 8.1.2.

Consider the sequence \(1, 2, 4, \ldots\text{.}\)
(a)
Which of the choices below reasonably continues this sequence of numbers?
  1. \(\displaystyle 7, 12, 24, \ldots\)
  2. \(\displaystyle 7, 11, 16, \ldots\)
  3. \(\displaystyle 8, 16, 32, \ldots\)
  4. \(\displaystyle 1, 2, 4, \ldots\)
  5. \(\displaystyle 7, 12, 20, \ldots\)
(b)
Where possible, find a formula that allows us to move from one term to the next one.

Remark 8.1.3.

As seen in the previous activity, having too few terms may prevent us from finding a unique way to continue creating a sequence of numbers. In fact, we need sufficiently many terms to uniquely continue a sequence of numbers (and how many terms is sufficient depends on which sequence of numbers you are trying to generate). Sometimes, we do not want to write out all of the terms needed to allow for this. Therefore, we will want to find short-hand notation that allows us to do so.

Definition 8.1.4.

A sequence is a list of real numbers. Let \(a_n\) denote the \(n\)th term in a sequence. We will use the notation \(\displaystyle \{a_n\}_{n=1}^\infty=a_1, a_2, \ldots, a_n, \ldots\text{.}\) A general formula that indicates how to explicitly find the \(n\)-th term of a sequence is the closed form of the sequence.

Activity 8.1.5.

Consider the sequence \(\displaystyle 1, \frac{1}{3}, \frac{1}{9}, \frac{1}{27}, \ldots\text{.}\) Which of the following choices gives a closed formula for this sequence? Select all that apply.
  1. \(\displaystyle \displaystyle \left\{\left(\frac{1}{3}\right)^{n-1}\right\}_{n=1}^\infty\)
  2. \(\displaystyle \displaystyle \left\{\left(\frac{1}{3}\right)^{n}\right\}_{n=1}^\infty\)
  3. \(\displaystyle \displaystyle \left\{\left(\frac{1}{3}\right)^{n-1}\right\}_{n=2}^\infty\)
  4. \(\displaystyle \displaystyle \left\{\left(\frac{1}{3}\right)^{n+1}\right\}_{n=0}^\infty\)
  5. \(\displaystyle \displaystyle \left\{\left(\frac{1}{3}\right)^{n}\right\}_{n=0}^\infty\)

Activity 8.1.6.

Let \(a_n\) be the \(n\)th term in the sequence \(\displaystyle \left\{\frac{n+1}{n}\right\}_{n=1}^\infty\text{.}\) Which of the following terms corresponds to the \(27^{th}\) term of this sequence?
  1. \(\displaystyle \frac{27}{26}\)
  2. \(\displaystyle \frac{26}{27}\)
  3. \(\displaystyle \frac{27}{28}\)
  4. \(\displaystyle \frac{28}{27}\)
  5. \(\displaystyle \frac{29}{28}\)

Activity 8.1.7.

Let \(a_n\) be the \(n\)th term in the sequence \(\displaystyle \left\{\frac{n+1}{n}\right\}_{n=2}^\infty\text{.}\) Which of the following terms corresponds to the \(27^{th}\) term of this sequence?
  1. \(\displaystyle \frac{27}{26}\)
  2. \(\displaystyle \frac{26}{27}\)
  3. \(\displaystyle \frac{27}{28}\)
  4. \(\displaystyle \frac{28}{27}\)
  5. \(\displaystyle \frac{29}{28}\)

Activity 8.1.8.

Let \(a_n\) be the \(n\)th term in the sequence \(\displaystyle 1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \ldots\text{.}\) Identify the \(81\)st term of this sequence.
  1. \(\displaystyle \frac{1}{79}\)
  2. \(\displaystyle \frac{1}{80}\)
  3. \(\displaystyle \frac{1}{81}\)
  4. \(\displaystyle \frac{1}{82}\)
  5. \(\displaystyle \frac{1}{83}\)

Activity 8.1.9.

Find a closed form for the sequence \(0, 3, 8, 15, 24, \ldots\text{.}\)

Activity 8.1.10.

Find a closed form for the sequence \(\displaystyle \frac{12}{1}, \frac{16}{2}, \frac{20}{3}, \frac{24}{4}, \frac{28}{5}, \ldots\text{.}\)

Activity 8.1.11.

Let \(a_n\) be the \(n\)th term in the sequence \(1, 1, 2, 3, 5, 8, \ldots\text{.}\) Find a formula for \(a_n\text{.}\)

Definition 8.1.12.

A sequence is recursive if the terms are defined as a function of previous terms (with the necessary initial terms provided).

Activity 8.1.13.

Consider the sequence defined by \(a_1=6\) and \(a_{k+1}=4a_k-7\) for \(k\geq 1\text{.}\) What are the first four terms?

Activity 8.1.14.

Consider the sequence \(2, 7, 22, 67, 202, \ldots\text{.}\) Which of the following offers the best recursive formula for this sequence?
  1. \(\displaystyle a_{n+1} = 3a_n+1\)
  2. \(a_1=2, a_k=3a_{k-1}+1\) for \(k>1\)
  3. \(a_1=2, a_2=7, a_k=3a_{k-1}+1\) for \(k>2\)

Activity 8.1.15.

Once more, consider the sequence \(1, 1, 2, 3, 5, 8, \ldots\) from Activity 8.1.11. Suppose \(a_1=1\) and \(a_2=1\text{.}\) Give a recursive formula for \(a_n\) for all \(n\geq 3\text{.}\)

Activity 8.1.16.

Give a recursive formula that generates the sequence \(1, 2, 4, 8, 16, 32, \ldots\text{.}\)

Activity 8.1.17.

(a)
Find the first 5 terms of the following sequence:
  • \(\displaystyle a_n=3 \cdot 2^{n}.\)
(b)
Find a closed form for the following sequence:
  • \(\displaystyle 4, 5, 8, 13, 20,\ldots\)
(c)
Find a recursive form for the following sequence:
  • \(\displaystyle -3, 2, 7, 12, 17,\ldots\)

Activity 8.1.18.

(a)
Find the first 5 terms of the following sequence:
  • \(\displaystyle a_n=5 \, n + 4.\)
(b)
Find a closed form for the following sequence:
  • \(\displaystyle 0, 1, 4, 9, 16,\ldots\)
(c)
Find a recursive form for the following sequence:
  • \(\displaystyle 2, -1, \frac{1}{2}, -\frac{1}{4}, \frac{1}{8},\ldots\)

Subsection 8.1.2 Videos

Figure 175. Video: Define and use explicit and recursive formulas for sequences
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