64 Questions
What is the defining characteristic of a geometric sequence?
Each term is obtained by multiplying the previous term by a constant ratio
How do geometric sequences differ from arithmetic sequences?
Geometric sequences involve multiplying terms, while arithmetic sequences involve adding or subtracting terms
Which formula represents the nth term of a geometric sequence?
$T_n = a \times r^{n-1}$
What type of growth or decay do geometric sequences exhibit?
Exponential growth or decay
What does the common ratio in a geometric sequence determine?
The multiplication factor between terms
In what situations can the geometric sequence formula be a powerful tool?
Financial calculations
What happens if the common ratio of a geometric sequence is greater than 1?
The sequence grows exponentially
Which term of a geometric sequence is represented by T_5?
$T_4$
What is the formula for calculating the nth term of a geometric sequence?
$T_n = a \times r^{n-1}$
What is the common ratio of the sequence 100, 50, 25, 12.5,...?
0.5
If the first term of a geometric sequence is 1 and the common ratio is 3, what is the nth term formula?
$1 \times 3^{(n-1)}$
In the sequence defined by variable-based terms p, 3p/2, 9p/3, what is the common ratio if 'p' equals 8?
1.5
What would be the next term in the sequence 5, -10, 20, -40,...?
-80
How does a geometric sequence play a role in modeling disease spread during a flu epidemic?
By representing how the epidemic spreads exponentially
What field of study can benefit from understanding geometric sequences when modeling compound interest?
Finance
For what purpose are geometric sequences particularly useful in real-world applications?
Modeling multiplicative growth or decay phenomena
What is the nth term formula for the geometric sequence with first term $a=1$ and common ratio $r=2$?
$T_n = 1 imes 2^{n-1}$
If the first term of a geometric sequence is 100 and the common ratio is 0.5, what is the formula for the nth term?
$T_n = 100 imes (0.5)^{n-1}$
If the first term of a geometric sequence is 6 and the common ratio is 1.5, what is the 5th term of the sequence?
13.5
What is the common ratio of the geometric sequence 5, -10, 20, -40,...?
-2
In the context of modeling disease spread, how can geometric sequences be used?
To predict the growth of the infected population over time
What is the common ratio of the geometric sequence 100, 50, 25, 12.5,...?
0.5
What is the next term in the sequence 5, -10, 20, -40,...?
-80
What is the common ratio of the variable-based sequence $p, 3p/2, 9p/3,...$ if $p=8$?
1.5
Which of the following fields can benefit from understanding geometric sequences when modeling compound interest?
Finance
What does the common ratio in a geometric sequence determine?
The type of growth or decay exhibited by the sequence
What is the formula for finding the nth term of a geometric sequence?
$T_n = a \times r^{(n-1)}$
What type of growth or decay do geometric sequences exhibit?
Exponential
If the first term 'a' of a geometric sequence is 2 and the common ratio 'r' is 5, what is the 4th term of the sequence?
320
What is the common ratio of the geometric sequence: 3, 9, 27, 81, ...?
3
How does a geometric sequence differ from an arithmetic sequence?
Geometric sequences have a constant product between terms.
Which characteristic distinguishes a geometric sequence from other types of sequences?
Multiplicative rate of change
In a geometric sequence with a first term 'a' and common ratio 'r', what does the common ratio determine?
The rate of change between consecutive terms
If a geometric sequence has a first term of 10 and a common ratio of 1/2, what is the 6th term of the sequence?
$\frac{5}{16}$
What happens if the common ratio of a geometric sequence is less than 1?
The terms in the sequence decrease.
Given a geometric sequence with 'a=3' and 'r=4', what is the value of 'n' if $T_n = 192$?
5
Which of the following is the correct formula for the $n^{th}$ term of a geometric sequence with first term $a$ and common ratio $r$?
$T_n = a imes r^{n-1}$
In the geometric sequence $5, -10, 20, -40,...$, what is the common ratio?
-2
If the first term of a geometric sequence is $p$ and the common ratio is $r$, what is the formula for the $n^{th}$ term of the sequence?
$T_n = p imes r^{n-1}$
In the variable-based geometric sequence $p, 3p/2, 9p/3,...$, if $p = 8$, what is the common ratio?
1.5
What is the common ratio of the geometric sequence $100, 50, 25, 12.5,...$?
0.5
In the context of modeling disease spread, how can geometric sequences be used?
To predict the growth of the population infected over time
What characteristic distinguishes a geometric sequence from other types of sequences?
The ratio between consecutive terms is constant
If the first term of a geometric sequence is 10 and the common ratio is $1/2$, what is the 6th term of the sequence?
0.625
If the first term of a geometric sequence is 3 and the common ratio is 4, what is the value of $n$ if the 4th term is 192?
6
What type of growth or decay do geometric sequences exhibit?
Exponential growth or decay
Which of the following fields can benefit from understanding geometric sequences when modeling compound interest?
Finance
If the first term of a geometric sequence is $a$ and the common ratio is $r$, which of the following expressions represents the $n$th term of the sequence?
$a(r^{n-1})$
If the first five terms of a geometric sequence are $3, 6, 12, 24, 48$, what is the common ratio?
2
What is the sum of the first $n$ terms of a geometric sequence with first term $a$ and common ratio $r$, if $r \neq 1$?
$a \cdot \frac{1 - r^n}{1 - r}$
If the first term of a geometric sequence is $5$ and the common ratio is $-2$, what is the sum of the first $10$ terms?
$-2555$
If the sum of the first $n$ terms of a geometric sequence is $\frac{a(1 - r^n)}{1 - r}$, and the sum of the first $(n+1)$ terms is $\frac{a(1 - r^{n+1})}{1 - r}$, what is the $(n+1)$th term?
$a(r^{n+1})$
If the first term of a geometric sequence is $12$ and the common ratio is $\frac{1}{3}$, what is the sum of the first $10$ terms?
$36$
If the first term of a geometric sequence is $1$ and the common ratio is $r$, what is the sum of the infinite series?
$\frac{1}{1 - r}$
What does the general formula for the nth term of a geometric sequence allow for?
Direct determination of any term within the sequence without all previous terms
In a geometric sequence, how do terms grow or decrease?
At a multiplicative rate
What type of growth or decay can result from a geometric sequence?
Exponential growth or decay
How is a geometric sequence formula different from an arithmetic sequence formula?
It encapsulates exponential growth or decay based on the common ratio
What characteristic makes geometric sequences useful in financial calculations and population studies?
Multiplicative rate of growth or decay
What does the common ratio in a geometric sequence determine?
The type of growth or decay exhibited
In the variable-based geometric sequence $p, 3p/2, 9p/3, ...$, what is the common ratio if $p = 8$?
1.5
Which of the following fields can benefit from understanding geometric sequences when modeling compound interest?
Finance
If the first term of a geometric sequence is $a$ and the common ratio is $r$, which of the following expressions represents the $n$th term of the sequence?
$a imes r^{n-1}$
How do geometric sequences play a role in modeling disease spread during a flu epidemic?
They are used to model the exponential growth of the number of infected individuals over time.
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