2.4 Sequences

• An arithmetic sequence is defined as a sequence where the difference between consecutive terms is constant
• In an arithmetic sequence, the common difference is constant
• A geometric sequence is defined as a sequence where each term is multiplied by a constant called the common ratio
• The common ratio in a geometric sequence helps predict the next term by multiplying it with the previous term
• The formula to find the nth term of an arithmetic sequence is T_n = a + (n - 1)d
• Match the definition with the formula:

  nth Term of Arithmetic Sequence ↔️ $T_{n} =$$a +$$(n - 1)d$

  Recursive Definition of Arithmetic Sequence ↔️ $T_{n} =$$T_{n - 1} +$$d$
• In a geometric sequence, each term is multiplied by a constant called the common ratio
• The common ratio in a geometric sequence helps predict the next term by multiplying it with the previous term
• What is the constant difference in an arithmetic sequence called?

  Common difference
• The formula to find the nth term of an arithmetic sequence is T_{n} = a + (n - 1)d
• What is a key advantage of the nth term formula over the recursive definition?

  Directly calculates any term
• The 10th term of the arithmetic sequence 2, 5, 8, 11... is29.
• The common difference in an arithmetic sequence is the difference between any two consecutive terms
• What is the value of \(d\) in the arithmetic sequence 2, 5, 8, 11...?

  3
• The 10th term of the arithmetic sequence 2, 5, 8, 11... is calculated as 29.
• What is the formula to find the nth term of a geometric sequence?

  T_{n} = a \cdot r^{n - 1}
• The 5th term of the geometric sequence 2, 6, 18, 54... is 162
• The 5th term of the geometric sequence 2, 6, 18, 54... is calculated as 162.
• What is the formula to find the sum of the first n</latex> terms of an arithmetic series?

  S_{n} = \frac{n}{2}(2a + (n - 1)d)
• Match the method with its advantage or disadvantage:

  Using the formula ↔️ Quick and efficient

  Summing individual terms ↔️ Time-consuming for large n
• What is the sum of the first 10 terms of the arithmetic sequence 3, 6, 9, ...?

  165
• The sum of the first 10 terms of the arithmetic sequence 3, 6, 9, ... is 165.
• An arithmetic sequence is a list of numbers where the difference between any two consecutive terms is constant
• What is the formula for the sum of the first $n$ terms of an arithmetic sequence?


  $S_{n} =$$\frac{n}{2}(2a + (n - 1)d)$
• $S_{n}$ represents the sum of the first n terms of an arithmetic sequence.
• What does $a$ represent in the formula for the sum of an arithmetic sequence?


  First term
• In the formula for the sum of an arithmetic sequence, n</latex> stands for the number of terms.
• What is the common difference $d$ in an arithmetic sequence?


  Constant difference between terms
• Using the formula for the sum of an arithmetic sequence is quick and efficient, especially for large values of n.
• Summing individual terms is time-consuming for arithmetic sequences with a large number of terms.
• What is the first term in the arithmetic sequence 3, 6, 9, ...?

  3
• In the arithmetic sequence 3, 6, 9, ..., the common difference d</latex> is 3.
• How many terms are in the sequence for calculating its sum in the example?

  10
• What is the defining characteristic of an arithmetic sequence?

  Constant difference between terms
• The constant difference between consecutive terms in an arithmetic sequence is called the common difference.
• The common difference in an arithmetic sequence is always constant.
• What type of sequence has a constant ratio between its terms?

  Geometric
• In an arithmetic sequence, knowing the common difference allows you to predict the next term.
• Match the sequence type with its defining characteristic:

  Arithmetic ↔️ Constant difference

  Geometric ↔️ Constant ratio

  Fibonacci ↔️ Sum of previous two terms
• What is the defining characteristic of a geometric sequence?

  Constant ratio between terms