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=1−ra1(1−rn)
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=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=a1+(n−1)d