CH 1: Geometric Sequences

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What is the defining characteristic of a geometric sequence?

Each term is added to the previous term

Each term is obtained by multiplying the previous term by a constant ratio (correct)

Each term is divided by the previous term

Each term is subtracted from the previous term

How do geometric sequences differ from arithmetic sequences?

Geometric sequences involve adding terms, while arithmetic sequences involve multiplying terms

Geometric sequences involve multiplying terms, while arithmetic sequences involve adding or subtracting terms (correct)

Both geometric and arithmetic sequences involve multiplying terms

Both geometric and arithmetic sequences involve adding terms

Which formula represents the nth term of a geometric sequence?

$T_n = \frac{a}{r^{n-1}}$

$T_n = a + r^{n-1}$

$T_n = a \times r^{n-1}$ (correct)

$T_n = a - r^{n-1}$

What type of growth or decay do geometric sequences exhibit?

Exponential growth or decay Signup and view all the answers

What does the common ratio in a geometric sequence determine?

The multiplication factor between terms Signup and view all the answers

In what situations can the geometric sequence formula be a powerful tool?

Financial calculations Signup and view all the answers

What happens if the common ratio of a geometric sequence is greater than 1?

The sequence grows exponentially Signup and view all the answers

Which term of a geometric sequence is represented by T_5?

$T_4$ Signup and view all the answers

What is the formula for calculating the nth term of a geometric sequence?

$T_n = a \times r^{n-1}$ Signup and view all the answers

What is the common ratio of the sequence 100, 50, 25, 12.5,...?

0.5 Signup and view all the answers

If the first term of a geometric sequence is 1 and the common ratio is 3, what is the nth term formula?

$1 \times 3^{(n-1)}$ Signup and view all the answers

In the sequence defined by variable-based terms p, 3p/2, 9p/3, what is the common ratio if 'p' equals 8?

1.5 Signup and view all the answers

What would be the next term in the sequence 5, -10, 20, -40,...?

-80 Signup and view all the answers

How does a geometric sequence play a role in modeling disease spread during a flu epidemic?

By representing how the epidemic spreads exponentially Signup and view all the answers

What field of study can benefit from understanding geometric sequences when modeling compound interest?

Finance Signup and view all the answers

For what purpose are geometric sequences particularly useful in real-world applications?

Modeling multiplicative growth or decay phenomena Signup and view all the answers

What is the nth term formula for the geometric sequence with first term $a=1$ and common ratio $r=2$?

$T_n = 1 imes 2^{n-1}$ Signup and view all the answers

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?

$T_n = 100 imes (0.5)^{n-1}$ Signup and view all the answers

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 Signup and view all the answers

What is the common ratio of the geometric sequence 5, -10, 20, -40,...?

-2 Signup and view all the answers

In the context of modeling disease spread, how can geometric sequences be used?

To predict the growth of the infected population over time Signup and view all the answers

What is the common ratio of the geometric sequence 100, 50, 25, 12.5,...?

0.5 Signup and view all the answers

What is the next term in the sequence 5, -10, 20, -40,...?

-80 Signup and view all the answers

What is the common ratio of the variable-based sequence $p, 3p/2, 9p/3,...$ if $p=8$?

1.5 Signup and view all the answers

Which of the following fields can benefit from understanding geometric sequences when modeling compound interest?

Finance Signup and view all the answers

What does the common ratio in a geometric sequence determine?

The type of growth or decay exhibited by the sequence Signup and view all the answers

What is the formula for finding the nth term of a geometric sequence?

$T_n = a \times r^{(n-1)}$ Signup and view all the answers

What type of growth or decay do geometric sequences exhibit?

Exponential Signup and view all the answers

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 Signup and view all the answers

What is the common ratio of the geometric sequence: 3, 9, 27, 81, ...?

3 Signup and view all the answers

How does a geometric sequence differ from an arithmetic sequence?

Geometric sequences have a constant product between terms. Signup and view all the answers

Which characteristic distinguishes a geometric sequence from other types of sequences?

Multiplicative rate of change Signup and view all the answers

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 Signup and view all the answers

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?

$\frac{5}{16}$ Signup and view all the answers

What happens if the common ratio of a geometric sequence is less than 1?

The terms in the sequence decrease. Signup and view all the answers

Given a geometric sequence with 'a=3' and 'r=4', what is the value of 'n' if $T_n = 192$?

5 Signup and view all the answers

Which of the following is the correct formula for the $n^{th}$ term of a geometric sequence with first term $a$ and common ratio $r$?

$T_n = a imes r^{n-1}$ Signup and view all the answers

In the geometric sequence $5, -10, 20, -40,...$, what is the common ratio?

-2 Signup and view all the answers

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?

$T_n = p imes r^{n-1}$ Signup and view all the answers

In the variable-based geometric sequence $p, 3p/2, 9p/3,...$, if $p = 8$, what is the common ratio?

1.5 Signup and view all the answers

What is the common ratio of the geometric sequence $100, 50, 25, 12.5,...$?

0.5 Signup and view all the answers

In the context of modeling disease spread, how can geometric sequences be used?

To predict the growth of the population infected over time Signup and view all the answers

What characteristic distinguishes a geometric sequence from other types of sequences?

The ratio between consecutive terms is constant Signup and view all the answers

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 Signup and view all the answers

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 Signup and view all the answers

What type of growth or decay do geometric sequences exhibit?

Exponential growth or decay Signup and view all the answers

Which of the following fields can benefit from understanding geometric sequences when modeling compound interest?

Finance Signup and view all the answers

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?

$a(r^{n-1})$ Signup and view all the answers

If the first five terms of a geometric sequence are $3, 6, 12, 24, 48$, what is the common ratio?

2 Signup and view all the answers

What is the sum of the first $n$ terms of a geometric sequence with first term $a$ and common ratio $r$, if $r \neq 1$?

$a \cdot \frac{1 - r^n}{1 - r}$ Signup and view all the answers

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?

$-2555$ Signup and view all the answers

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?

$a(r^{n+1})$ Signup and view all the answers

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?

$36$ Signup and view all the answers

If the first term of a geometric sequence is $1$ and the common ratio is $r$, what is the sum of the infinite series?

$\frac{1}{1 - r}$ Signup and view all the answers

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 Signup and view all the answers

In a geometric sequence, how do terms grow or decrease?

At a multiplicative rate Signup and view all the answers

What type of growth or decay can result from a geometric sequence?

Exponential growth or decay Signup and view all the answers

How is a geometric sequence formula different from an arithmetic sequence formula?

It encapsulates exponential growth or decay based on the common ratio Signup and view all the answers

What characteristic makes geometric sequences useful in financial calculations and population studies?

Multiplicative rate of growth or decay Signup and view all the answers

What does the common ratio in a geometric sequence determine?

The type of growth or decay exhibited Signup and view all the answers

In the variable-based geometric sequence $p, 3p/2, 9p/3, ...$, what is the common ratio if $p = 8$?

1.5 Signup and view all the answers

Which of the following fields can benefit from understanding geometric sequences when modeling compound interest?

Finance Signup and view all the answers

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?

$a imes r^{n-1}$ Signup and view all the answers

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. Signup and view all the answers

CH 1: Geometric Sequences

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Questions and Answers

What is the defining characteristic of a geometric sequence?

How do geometric sequences differ from arithmetic sequences?

Which formula represents the nth term of a geometric sequence?

What type of growth or decay do geometric sequences exhibit?

What does the common ratio in a geometric sequence determine?

In what situations can the geometric sequence formula be a powerful tool?

What happens if the common ratio of a geometric sequence is greater than 1?

Which term of a geometric sequence is represented by T_5?

What is the formula for calculating the nth term of a geometric sequence?

What is the common ratio of the sequence 100, 50, 25, 12.5,...?

If the first term of a geometric sequence is 1 and the common ratio is 3, what is the nth term formula?

In the sequence defined by variable-based terms p, 3p/2, 9p/3, what is the common ratio if 'p' equals 8?

What would be the next term in the sequence 5, -10, 20, -40,...?

How does a geometric sequence play a role in modeling disease spread during a flu epidemic?

What field of study can benefit from understanding geometric sequences when modeling compound interest?

For what purpose are geometric sequences particularly useful in real-world applications?

What is the nth term formula for the geometric sequence with first term $a=1$ and common ratio $r=2$?

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?

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?

What is the common ratio of the geometric sequence 5, -10, 20, -40,...?

In the context of modeling disease spread, how can geometric sequences be used?

What is the common ratio of the geometric sequence 100, 50, 25, 12.5,...?

What is the next term in the sequence 5, -10, 20, -40,...?

What is the common ratio of the variable-based sequence $p, 3p/2, 9p/3,...$ if $p=8$?

Which of the following fields can benefit from understanding geometric sequences when modeling compound interest?

What does the common ratio in a geometric sequence determine?

What is the formula for finding the nth term of a geometric sequence?

What type of growth or decay do geometric sequences exhibit?

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?

What is the common ratio of the geometric sequence: 3, 9, 27, 81, ...?

How does a geometric sequence differ from an arithmetic sequence?

Which characteristic distinguishes a geometric sequence from other types of sequences?

In a geometric sequence with a first term 'a' and common ratio 'r', what does the common ratio determine?

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?

What happens if the common ratio of a geometric sequence is less than 1?

Given a geometric sequence with 'a=3' and 'r=4', what is the value of 'n' if $T_n = 192$?

Which of the following is the correct formula for the $n^{th}$ term of a geometric sequence with first term $a$ and common ratio $r$?

In the geometric sequence $5, -10, 20, -40,...$, what is the common ratio?

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?

In the variable-based geometric sequence $p, 3p/2, 9p/3,...$, if $p = 8$, what is the common ratio?

What is the common ratio of the geometric sequence $100, 50, 25, 12.5,...$?

In the context of modeling disease spread, how can geometric sequences be used?

What characteristic distinguishes a geometric sequence from other types of sequences?

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?

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?

What type of growth or decay do geometric sequences exhibit?

Which of the following fields can benefit from understanding geometric sequences when modeling compound interest?

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?

If the first five terms of a geometric sequence are $3, 6, 12, 24, 48$, what is the common ratio?

What is the sum of the first $n$ terms of a geometric sequence with first term $a$ and common ratio $r$, if $r \neq 1$?

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?

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?

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?

If the first term of a geometric sequence is $1$ and the common ratio is $r$, what is the sum of the infinite series?

What does the general formula for the nth term of a geometric sequence allow for?

In a geometric sequence, how do terms grow or decrease?

What type of growth or decay can result from a geometric sequence?

How is a geometric sequence formula different from an arithmetic sequence formula?

What characteristic makes geometric sequences useful in financial calculations and population studies?

What does the common ratio in a geometric sequence determine?

In the variable-based geometric sequence $p, 3p/2, 9p/3, ...$, what is the common ratio if $p = 8$?

Which of the following fields can benefit from understanding geometric sequences when modeling compound interest?

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?

How do geometric sequences play a role in modeling disease spread during a flu epidemic?

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