CH 1: Geometric Sequences
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Questions and Answers

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?

<p>Exponential growth or decay (A)</p> Signup and view all the answers

What does the common ratio in a geometric sequence determine?

<p>The multiplication factor between terms (C)</p> Signup and view all the answers

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

<p>Financial calculations (D)</p> Signup and view all the answers

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

<p>The sequence grows exponentially (D)</p> Signup and view all the answers

Which term of a geometric sequence is represented by T_5?

<p>$T_4$ (B)</p> Signup and view all the answers

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

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

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

<p>0.5 (D)</p> 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?

<p>$1 \times 3^{(n-1)}$ (A)</p> 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?

<p>1.5 (B)</p> Signup and view all the answers

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

<p>-80 (B)</p> Signup and view all the answers

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

<p>By representing how the epidemic spreads exponentially (A)</p> Signup and view all the answers

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

<p>Finance (D)</p> Signup and view all the answers

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

<p>Modeling multiplicative growth or decay phenomena (A)</p> Signup and view all the answers

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

<p>$T_n = 1 imes 2^{n-1}$ (B)</p> 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?

<p>$T_n = 100 imes (0.5)^{n-1}$ (A)</p> 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?

<p>13.5 (A)</p> Signup and view all the answers

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

<p>-2 (C)</p> Signup and view all the answers

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

<p>To predict the growth of the infected population over time (B)</p> Signup and view all the answers

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

<p>0.5 (B)</p> Signup and view all the answers

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

<p>-80 (C)</p> Signup and view all the answers

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

<p>1.5 (C)</p> Signup and view all the answers

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

<p>Finance (A)</p> Signup and view all the answers

What does the common ratio in a geometric sequence determine?

<p>The type of growth or decay exhibited by the sequence (D)</p> Signup and view all the answers

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

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

What type of growth or decay do geometric sequences exhibit?

<p>Exponential (D)</p> 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?

<p>320 (B)</p> Signup and view all the answers

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

<p>3 (A)</p> Signup and view all the answers

How does a geometric sequence differ from an arithmetic sequence?

<p>Geometric sequences have a constant product between terms. (C)</p> Signup and view all the answers

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

<p>Multiplicative rate of change (C)</p> 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?

<p>The rate of change between consecutive terms (A)</p> 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?

<p>$\frac{5}{16}$ (D)</p> Signup and view all the answers

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

<p>The terms in the sequence decrease. (C)</p> Signup and view all the answers

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

<p>5 (A)</p> 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 $r$?

<p>$T_n = a imes r^{n-1}$ (D)</p> Signup and view all the answers

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

<p>-2 (A)</p> 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?

<p>$T_n = p imes r^{n-1}$ (C)</p> 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?

<p>1.5 (A)</p> Signup and view all the answers

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

<p>0.5 (C)</p> Signup and view all the answers

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

<p>To predict the growth of the population infected over time (D)</p> Signup and view all the answers

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

<p>The ratio between consecutive terms is constant (A)</p> 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?

<p>0.625 (B)</p> 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?

<p>6 (C)</p> Signup and view all the answers

What type of growth or decay do geometric sequences exhibit?

<p>Exponential growth or decay (C)</p> Signup and view all the answers

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

<p>Finance (A)</p> 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?

<p>$a(r^{n-1})$ (D)</p> 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?

<p>2 (C)</p> 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 $r \neq 1$?

<p>$a \cdot \frac{1 - r^n}{1 - r}$ (B)</p> 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?

<p>$-2555$ (B)</p> 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?

<p>$a(r^{n+1})$ (A)</p> 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?

<p>$36$ (D)</p> 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?

<p>$\frac{1}{1 - r}$ (D)</p> Signup and view all the answers

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

<p>Direct determination of any term within the sequence without all previous terms (D)</p> Signup and view all the answers

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

<p>At a multiplicative rate (B)</p> Signup and view all the answers

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

<p>Exponential growth or decay (C)</p> Signup and view all the answers

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

<p>It encapsulates exponential growth or decay based on the common ratio (B)</p> Signup and view all the answers

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

<p>Multiplicative rate of growth or decay (B)</p> Signup and view all the answers

What does the common ratio in a geometric sequence determine?

<p>The type of growth or decay exhibited (B)</p> Signup and view all the answers

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

<p>1.5 (A)</p> Signup and view all the answers

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

<p>Finance (C)</p> 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?

<p>$a imes r^{n-1}$ (C)</p> Signup and view all the answers

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

<p>They are used to model the exponential growth of the number of infected individuals over time. (C)</p> Signup and view all the answers

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