5.1 Sequences and Series

• What is a sequence in mathematics?

  An ordered list of terms
• In an arithmetic sequence, terms increase or decrease by a constant difference.
• In a geometric sequence, terms increase or decrease by a constant ratio
• What does the common difference represent in an arithmetic sequence?

  The constant increase or decrease between terms
• The common ratio in a geometric sequence is denoted by the letter r
• What is the general term formula for an arithmetic sequence?

  a_{n} = a_{1} + (n -1)d</latex>
• The general term formula for a geometric sequence is $a_{n} =$$a_{1} \cdot r^{n - 1}$.
• Match the sequence type with its description:

  Arithmetic ↔️ Constant difference

  Geometric ↔️ Constant ratio
• What is a series in mathematics?

  The sum of terms in a sequence
• The sum of an arithmetic series is given by S_{n} = \frac{n}{2}(2a + (n - 1)d)</latex>, where $n$ is the number of terms
• What is the sum of a finite geometric series?

  $\frac{a(1 - r^{n})}{1 - r}$
• An arithmetic series diverges if the common difference is not zero.
• Under what condition does an infinite geometric series converge?

  $|r| < 1$
• Order the steps to calculate the general term of an arithmetic sequence:

  1️⃣ Identify the first term $a_{1}$

  2️⃣ Determine the common difference $d$

  3️⃣ Substitute $a_{1}$, $d$, and $n$ into the formula $a_{n} =$$a_{1} +$$(n - 1)d$
• Geometric sequences grow exponentially, while arithmetic sequences grow linearly.
• What is the purpose of sigma notation?

  Express the sum of a series
• The summation symbol in sigma notation is represented by the letter \Sigma
• Match the components of sigma notation with their descriptions:

  \Sigma ↔️ Summation symbol

  Index ↔️ Variable incremented in the sum

  Lower bound ↔️ Starting value of the index
• What does sigma notation express in a compact way?

  Sum of a series
• In sigma notation, the index is represented by the variable i
• The lower bound in sigma notation indicates the starting value of the index.
• Match the component of sigma notation with its description:

  $\Sigma$ ↔️ Summation symbol

  Index ↔️ Variable indicating terms to sum

  Term expression ↔️ Formula to calculate terms
• Order the steps to evaluate the following sigma notation: $\sum_{i = 1}^{5} i$


  1️⃣ Start with $i =$$1$

  2️⃣ Calculate the term: $i =$$1$

  3️⃣ Increase the index by 1: $i =$$2$

  4️⃣ Repeat until $i =$$5$

  5️⃣ Add all the calculated terms
• What is a sequence defined as?

  Ordered list of numbers
• An arithmetic sequence has a constant common difference
• The general term of a geometric sequence is a_{n} = a_{1} \cdot r^{n - 1}</latex>
• What is a series defined as?

  Sum of terms in a sequence
• The general term of an arithmetic sequence is a_{n} = a_{1} + (n - 1)d</latex>
• The general term of a geometric sequence is $a_{n} =$$a_{1} \cdot r^{n - 1}$
• Match the type of sequence with its defining characteristic:

  Arithmetic ↔️ Common difference

  Geometric ↔️ Common ratio
• What is the formula for the partial sum of an arithmetic sequence?

  S_{n} = \frac{n}{2}(2a + (n - 1)d)</latex>
• What is a partial sum for an arithmetic sequence?

  $S_{n} =$$\frac{n}{2}(2a + (n - 1)d)$
• For the sequence $2, 4, 6, 8, 10$, the partial sum of the first 5 terms is 30
• What is the formula for the partial sum of a geometric sequence?

  S_{n} = \frac{a(1 - r^{n})}{1 - r}</latex>
• The partial sum of a geometric sequence requires $r \neq 1$.
• An infinite geometric series converges when $|r| < 1$
• What is the formula for the sum of an infinite geometric series when it converges?

  $S_{\infty} =$$\frac{a_{1}}{1 - r}$
• An infinite geometric series diverges when $|r| \geq 1$.
• What are the two components required for a recurrence relation?

  Initial condition and formula
• In a recurrence relation, the formula is typically $a_{n} =$$f(a_{n - 1})$.preceding