2.3 Sequences

• In an arithmetic sequence, each term is obtained by adding a constant difference
• The formula to find the $n$-th term of an arithmetic sequence is a_{n} = a_{1} + (n - 1)d</latex>, where $d$ represents the common difference
• Steps to determine the common difference in an arithmetic sequence

  1️⃣ Identify two consecutive terms

  2️⃣ Subtract the earlier term from the later term

  3️⃣ The result is the common difference
• The sum of an arithmetic sequence can be calculated using the formula S_{n} = \frac{n}{2} (a_{1} + a_{n})</latex>, where $a_{1}$ is the first term
• To find the sum of the first 10 terms of the sequence 3, 6, 9, ..., 30, the result is 165.

  True
• The formula to find the $n$-th term of an arithmetic sequence is $a_{n} =$$a_{1} +$$(n - 1)d$, where d</latex> is the common difference
• In an arithmetic sequence, the common difference is added to each term to get the next term.
• What is the 15th term of the arithmetic sequence 3, 7, 11, 15, ...?

  59
• What is the formula to calculate the sum of the first n terms of an arithmetic sequence?

  $S_{n} =$$\frac{n}{2} (a_{1} +$$a_{n})$
• The sum of the first 10 terms of the arithmetic sequence 3, 6, 9, 12, ..., 30 is 165.


  True
• The common ratio in a geometric sequence is found by dividing any term by its preceding term.
• The 8th term of the geometric sequence $a_{n} =$$2 \cdot 3^{n - 1}$ is 4374.


  True
• What is the common ratio in the geometric sequence 4, 12, 36, 108, ...?

  3
• What is the sum of the first 5 terms of the geometric sequence 2, 4, 8, 16, 32?

  62
• What is the common difference in the arithmetic sequence 2, 4, 6, 8, 10?

  2
• What is the 15th term of the arithmetic sequence 3, 7, 11, 15, ...?

  59
• The common difference in an arithmetic sequence is the value added to each term to obtain the next term.


  True
• The common difference in an arithmetic sequence is the constant value added to each term to obtain the next term.


  True
• What does $a_{n}$ represent in the formula S_{n} = \frac{n}{2}(a_{1} + a_{n})</latex>?


  $n$-th term
• In the formula $a_{n} =$$a_{1} \cdot r^{n - 1}$, $n$ represents the term number.
• The common ratio in a geometric sequence is the constant factor by which each term is multiplied to obtain the next term.


  True
• A geometric sequence is obtained by multiplying each term by a common ratio.


  True
• Match the sequence type with its characteristic:

  Arithmetic Sequence ↔️ Common difference

  Geometric Sequence ↔️ Common ratio
• What does $a_{1}$ represent in the sum of a geometric sequence formula?


  The first term
• Name three real-life situations where sequences are used for modeling.

  Financial planning, population growth, resource management
• Sequences can be used to model loan repayments in financial planning.


  True
• In a geometric sequence, each term is obtained by multiplying the previous term by a constant ratio.


  True
• In the arithmetic sequence formula, $a_{1}$ represents the first term.


  True
• The common difference in an arithmetic sequence is the constant value added to each term to obtain the next term
• What does $S_{n}$ represent in the formula for the sum of an arithmetic sequence?


  Sum of first n terms
• What is a sequence in mathematics?

  Ordered list of numbers
• The common difference in an arithmetic sequence is the constant value added to each term to get the next term.


  True
• The formula to find the n</latex>-th term of an arithmetic sequence is <latex>a_{n} = a_{1} + (n - 1)d
• The common difference in an arithmetic sequence is found by subtracting any term from its succeeding term.
• In the formula for the sum of an arithmetic sequence, $S_{n}$ represents the sum of the first n terms.
• In a geometric sequence, each term is obtained by multiplying the previous term by a constant ratio.
• What is the common ratio in the geometric sequence 4, 12, 36, 108, ...?

  3
• In a geometric sequence, the common ratio is found by dividing any term by its preceding
• The sum of a geometric sequence can be calculated using the formula S_{n} = \frac{a_{1}(1 - r^{n})}{1 - r}</latex>, where $r$ is the common ratio
• A sequence in mathematics is an ordered list of numbers or objects that follow a specific pattern or rule.


  True