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Chapter 3: Problem 1

Decide which of the following sequences are geometric progressions. For those sequences that are of this type, write down their geometric ratios. (a) \(3,6,12,24, \ldots\) (b) \(5,10,15,20, \ldots\) (c) \(1,-3,9,-27, \ldots\) (d) \(8,4,2,1,1 / 2, \ldots\) (e) \(500,500(1.07), 500(1.07)^{2}, \ldots\)

Short Answer

Expert verified
(a), (c), (d), (e) are geometric progressions with ratios 2, -3, 0.5, 1.07 respectively.

Step by step solution

01

Define a geometric progression

A sequence is a geometric progression if each term after the first is the previous term multiplied by a constant ratio, denoted as \( r \).
02

Check sequence (a) for geometric progression

Consider sequence \(3, 6, 12, 24, \ldots\)\. Calculate the ratio \( r \) for successive terms: \[ r = \frac{6}{3} = 2, \frac{12}{6} = 2, \frac{24}{12} = 2 \] Since the ratio is constant, it is a geometric progression with \( r = 2 \).
03

Check sequence (b) for geometric progression

Consider sequence \(5, 10, 15, 20, \ldots\)\. Calculate the ratio \( r \) for successive terms: \[ r = \frac{10}{5} = 2, \frac{15}{10} = 1.5, \frac{20}{15} = \frac{4}{3} \] Since the ratio is not constant, it is not a geometric progression.
04

Check sequence (c) for geometric progression

Consider sequence \(1, -3, 9, -27, \ldots\)\. Calculate the ratio \( r \) for successive terms: \[ r = \frac{-3}{1} = -3, \frac{9}{-3} = -3, \frac{-27}{9} = -3 \] Since the ratio is constant, it is a geometric progression with \( r = -3 \).
05

Check sequence (d) for geometric progression

Consider sequence \(8, 4, 2, 1, \frac{1}{2}, \ldots\)\. Calculate the ratio \( r \) for successive terms: \[ r = \frac{4}{8} = \frac{1}{2}, \frac{2}{4} = \frac{1}{2}, \frac{1}{2} = \frac{1}{2} \] Since the ratio is constant, it is a geometric progression with \( r = \frac{1}{2} \).
06

Check sequence (e) for geometric progression

Consider sequence \(500, 500(1.07), 500(1.07)^{2}, \ldots\)\. Calculate the ratio \( r \) for successive terms: \[ r = \frac{500(1.07)}{500} = 1.07, \frac{500(1.07)^{2}}{500(1.07)} = 1.07 \] Since the ratio is constant, it is a geometric progression with \( r = 1.07 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Ratios
Ratios are simple math tools that express the relationship between two numbers. In a geometric progression, the ratio remains constant across all terms in the sequence. For example, in the sequence 3, 6, 12, 24,..., we calculate the ratio by dividing each term by the preceding one: \(r = \frac{6}{3} = 2, \frac{12}{6} = 2, \frac{24}{12} = 2\). Notice that the ratio (2 in this case) does not change. This consistency tells us that the sequence is a geometric progression. Understanding and calculating ratios are crucial for identifying geometric sequences.
  • The ratio (r) is the factor used to multiply the previous term to get the next term.
  • If the ratio varies, the sequence is not geometric.
Sequence Analysis
Analyzing sequences involves checking if the given sequence follows a specific pattern. In the context of geometric progressions, this pattern is determined by a constant ratio between terms.
Consider the sequence 1, -3, 9, -27,...
Let鈥檚 calculate the ratios: \( r = \frac{-3}{1} = -3, \frac{9}{-3} = -3, \frac{-27}{9} = -3\). Since the ratio is consistently -3, the sequence is a geometric progression.
  • Analyze each sequence by dividing terms to find potential ratios.
  • If a constant ratio exists, the sequence is geometric.
Effective sequence analysis enables you to categorize sequences correctly and apply appropriate mathematical tools.
Mathematical Series
A series is the sum of the terms of a sequence. When dealing with geometric progressions, we often need to understand the sum of terms to solve various problems. For a geometric series, the sum of the first n terms (denoted as S_n) can be calculated using the formula: \[ S_n = a \frac{1 - r^n}{1 - r} \] where 'a' is the first term and 'r' is the common ratio.
  • This formula helps in quickly finding the sum without manually adding terms.
  • It鈥檚 crucial to know if 鈥榬鈥� is greater or less than 1 for correct application.
Understanding this concept opens up various possibilities in solving mathematical problems involving the sums of geometric progressions.
Constant Ratio
The backbone of any geometric progression is its constant ratio. This is what differentiates it from other types of sequences. The beauty of geometric progressions lies in their predictability; knowing the first term and the ratio allows you to determine any term in the sequence.
Let鈥檚 revisit a scenario: Given the sequence 500, 500(1.07), 500(1.07)^2,...
The ratio is consistently calculated as \[ r = \frac{500(1.07)}{500} = 1.07, \frac{500(1.07)^{2}}{500(1.07)}= 1.07 \]. With a constant ratio of 1.07, this sequence is confirmed to be a geometric progression.
  • A constant ratio helps in identifying geometric sequences.
  • It also aids in constructing the sequence or series without ambiguity.
Mastering the concept of a constant ratio is fundamental in learning and applying geometric progressions accurately.

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Most popular questions from this chapter

An investment project requires an initial outlay of \(\$ 8000\) and will produce a return of \(\$ 17000\) at the end of 5 years. Use the (a) net present value (b) internal rate of return methods to decide whether this is worthwhile if the capital could be invested elsewhere at \(15 \%\) compounded annually.

(Excel) The sum of \(\$ 100\) is invested at \(12 \%\) interest for 20 years. Tabulate the value of the investment at the end of each year, if the interest is compounded (a) annually (b) quarterly (c) monthly (d) continuously Draw graphs of these values on the same diagram. Comment briefly on any similarities and differences between these graphs.

The turnover of a leading supermarket chain, \(A\), is currently \(\$ 560\) million and is expected to increase at a constant rate of \(1.5 \%\) a year. Its nearest rival, supermarket \(B\), has a current turnover of \(\$ 480\) million and plans to increase this at a constant rate of \(3.4 \%\) a year. After how many years will supermarket B overtake supermarket \(A\) ?

A person requests an immediate bank overdraft of \(\$ 2000\). The bank generously agrees to this, but insists that it should be repaid by 12 monthly instalments and charges \(1 \%\) interest every month on the outstanding debt. Determine the monthly repayment.

A shop sells books at ' \(20 \%\) below the recommended retail price (r.r.p.)'. If it sells a book for \(拢 12.40\) find (a) the r.r.p. (b) the cost of the book after a further reduction of \(15 \%\) in a sale (c) the overall percentage discount obtained by buying the book from the shop in the sale compared with the manufacturer's r.r.p.
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