3.5 D: Sequences and Series

Cards (103)

  • What is a sequence in mathematics?
    Ordered list of numbers
  • What is the definition of a sequence in mathematics?
    Ordered list of numbers
  • A series is the sum of the terms in a sequence.
    True
  • An arithmetic sequence has a constant difference between consecutive terms.
    True
  • What is the common property of a geometric sequence?
    Constant ratio
  • What does the variable \( a_1 \) represent in the formula for the nth term of an arithmetic sequence?
    First term
  • An arithmetic sequence has a constant difference
  • The formula for the nth term of an arithmetic sequence is a_n
  • What is the key difference between arithmetic and geometric sequences?
    Constant difference vs. constant ratio
  • An arithmetic series is the sum of terms in a sequence with a constant difference
  • What is the purpose of the formula for the nth term in an arithmetic sequence?
    Find the value of the nth term
  • What does the variable 'd' represent in the formula for the nth term of an arithmetic sequence?
    Common difference
  • The sum of the first \( n \) terms in a geometric series is calculated using the formula \( S_n = \frac{a_1(1 - r^n)}{1 - r} \)

    True
  • What is the formula for the nth term of an arithmetic sequence?
    an=a_{n} =a1+ a_{1} +(n−1)d (n - 1)d
  • Summarize the key differences between arithmetic and geometric sequences.
    1️⃣ Constant difference vs. constant ratio
    2️⃣ Common difference \( d \) vs. common ratio \( r \)
    3️⃣ an=a_{n} =a1+ a_{1} +(n−1)d (n - 1)d vs. an=a_{n} =a1⋅r(n−1) a_{1} \cdot r^{(n - 1)}
    4️⃣ Example: 2, 4, 6, 8, ... vs. 3, 6, 12, 24, ...
  • What is the value of the common difference \( d \) in the arithmetic sequence 2, 4, 6, 8, ...?
    2
  • The nth term formula helps find specific terms in a geometric sequence
  • Match the variable with its meaning in the geometric sequence formulas:
    a_1 ↔️ First term
    n ↔️ Term number
    r ↔️ Common ratio
  • The formula for the sum of \( n \) terms in a geometric series is Sn=S_{n} =a1(1−rn)1−r \frac{a_{1}(1 - r^{n})}{1 - r}
  • The 10th term of the arithmetic sequence 3, 7, 11, 15, ... is 39
  • What is the 8th term of the geometric sequence 2, 6, 18, 54, ...?
    a8=a_{8} =4374 4374
  • The sum of the first n terms of a geometric series is given by \( S_n = \frac{a_1(1 - r^n)}{1 - r} \)

    True
  • The sum of the first 6 terms of the geometric sequence 5, 10, 20, 40, ... is 315

    True
  • Match the sigma notation component with its description:
    Σ ↔️ Represents the sum
    i ↔️ Variable taking values from a to b
    f(i) ↔️ Expression or term being summed
  • An infinite geometric series has terms forming a geometric sequence with a common ratio.
  • What is the formula for the sum \( S \) of an infinite geometric series when \( |r| < 1 \)?
    S = \frac{a_1}{1 - r}
  • An infinite geometric series converges if the absolute value of the common ratio is less than 1.

    True
  • An infinite geometric series continues indefinitely, rather than having a finite number of terms
  • The formula for the nth term of an arithmetic sequence is \( a_n = a_1 + (n-1) d
  • Arithmetic sequences have a constant difference, while geometric sequences have a constant ratio.

    True
  • The common difference in an arithmetic sequence is denoted by \( d \).

    True
  • The formula for the nth term of an arithmetic sequence is a_n
  • The sum of the first \( n \) terms in an arithmetic series is calculated using the formula \( S_n = \frac{n}{2}(a_1 + a_n) \), where \( a_n \) is the last term.
  • What is the difference between a sequence and a series?
    Sequence is ordered, series is sum
  • Geometric sequences have a constant ratio between consecutive terms.

    True
  • The sum of the first \( n \) terms in an arithmetic series is calculated using the formula \( S_n = \frac{n}{2}(a_1 + a_n) \), where \( a_n \) is the nth term.
  • What is the sum of the first 5 terms in the arithmetic sequence 2, 4, 6, 8, ...?
    30
  • What is the formula for the nth term of a geometric sequence?
    a_n = a_1 \cdot r^{(n-1)}</latex>
  • Steps to find the 4th term and the sum of the first 4 terms in the geometric sequence 2, 6, 18, ...
    1️⃣ Identify a_1, r, and n
    2️⃣ Calculate the 4th term using the nth term formula
    3️⃣ Calculate the sum of the first 4 terms using the sum formula
  • The nth term formula for an arithmetic sequence is an=a_{n} =a1+ a_{1} +(n−1)d (n - 1)d

    True