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Chapter 18: Problem 17

Determine whether or not the sum is geometric. Assume \("+\cdots\) indicates that the established pattern continues. If the sum is geometric, identify " \(a\) " and " \(r\) ". $$ p+p^{3}+p^{5}+p^{7}+p^{9} $$

Short Answer

Expert verified
The sum \(p+p^{3}+p^{5}+p^{7}+p^{9}\) is a geometric series with the first term 'a' being \(p\) and the common ratio 'r' being \(p^{2}\).

Step by step solution

01

Arranging the Sum

First arrange the sum that matches a potential geometric series. For the sum given: \(p+p^{3}+p^{5}+p^{7}+p^{9}\), we can rewrite it as \(p + p^3 + p^5 + p^7 + p^9 = p(1 + p^{2} + p^{4}+ p^{6} + p^{8})\), as this looks more like a typical geometric series with specifying weights.
02

Identifying whether it is a Geometric Series

Next, determine if it is a geometric series. A geometric series has the condition that each term is the previous term multiplied by a constant. Here the terms inside the bracket are: \(1, p^{2}, p^{4}, p^{6}, p^{8}\). If this series is geometric, the ratio between each successive term should be constant. Dividing each term by the previous one, the ratios are \(p^{2}, p^{2}, p^{2}, p^{2}\) respectively. This indicates that the sum inside the bracket is a geometric series.
03

Identifying the first term 'a' and the common ratio 'r'

Now, determine the first term \(a\) and the common ratio \(r\) of the series. Given the general form of a geometric series is \(a + ar + ar^2 + ar^3\cdots\), we can see here, 'a' is 1 and the common ratio 'r' is \(p^{2}\). These are the values for 'a' and 'r' of the sum inside the bracket. As for the entire sum, the first term 'a' is \(p\) and the common ratio is \(p^{2}\).

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Key Concepts

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

Geometric Sequence
A geometric sequence is a series of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number. This fixed number is known as the common ratio. Imagine you have a set of numbers where each one is generated from the one before it by a simple multiplication. This kind of sequence can grow or shrink very rapidly depending on whether the common ratio is greater or less than one. Here鈥檚 how it works:
  • Start with the first term, known as 'a'.
  • Multiply 'a' by the common ratio 'r' to get the next term.
  • Continue multiplying each new term by 'r' to find the subsequent terms.
Geometric sequences are found everywhere in nature and science, such as in the growth of populations or radioactive decay.
Common Ratio
The common ratio in a geometric sequence is the key factor that defines its pattern. It鈥檚 the number you multiply by to get from one term to the next. In our exercise, every term inside the bracket after the first was achieved by multiplying by the common ratio, which in this case is \(p^2\).
  • A common ratio greater than 1 will make your sequence grow.
  • When the common ratio is between 0 and 1, the sequence shrinks.
  • If the common ratio is negative, the sequence will alternate in sign.
Understanding the common ratio enables you to predict the future terms of the sequence effectively.
First Term
The first term of a sequence is like the starting point of a journey. In a geometric sequence, it is denoted by 'a'. This term is crucial since every subsequent term is generated from here using the common ratio. For example, in our solution, the first term of the sequence was determined to be 1 for the terms inside the bracket, making each future calculation relative to this initial term.
Similarly, looking at the entire sequence, the first term 'a' is \(p\).
  • This single term provides the base for constructing the entire sequence.
  • Often, changing the first term will affect the whole sequence pattern.
Knowing the first term is essential for defining and continuing a sequence.
Patterns in Series
Series often exhibit particular patterns that help us understand the structure and predictability of sequences. A geometric series, like the one identified in the exercise, clearly exhibits such regularity by consistently following a set multiplication pattern defined by the common ratio.
  • In our case, adding each new term involves a consistent multiplication by \(p^2\).
  • The pattern repetition allows for easier computation of the sum of the series.
  • By analyzing these patterns, you can extrapolate beyond the visible series and predict subsequent terms.
Recognizing patterns in series is not only crucial for solving mathematical problems but also offers insight into various natural and scientific phenomena.

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Most popular questions from this chapter

Determine whether or not the sum is geometric. Assume \("+\cdots\) indicates that the established pattern continues. If the sum is geometric, identify " \(a\) " and " \(r\) ". $$ \frac{2}{3}+\frac{2}{9}+\frac{2}{27}+\cdots+\frac{2}{6561} $$

Give an example of each of the following. (a) An in nite series that converges and whose partial sums are always increasing (b) An in nite series that converges and whose partial sums oscillate around the sum of the series (c) An in nite series that diverges although its terms approach zero (d) An in nite series that diverges but whose partial sums do not grow without bound

People who have slow metabolism due to a malfunctioning thyroid can take thyroid medication to alleviate their condition. For example, the boxer Muhammad Ali took Thyrolar 3, which is 3 grains of thyroid medication, every day. The amount of the drug in the bloodstream decays exponentially with time. The half-life of Thyrolar is 1 week. (a) Suppose one 3 -grain pill of Thyrolar is taken. Write an equation for the amount of the drug in the bloodstream \(t\) days after it has been taken. (Hint: In part (a) you are dealing with one 3 -grain pill of Thyrolar. Knowing the half-life of Thyrolar, you are asked to come up with a decay equation. This part of the problem has nothing to do with geometric sums.) (b) Suppose that Ali starts with none of the drug in his bloodstream. If he takes 3 grains of Thyrolar every day for ve days, how much Thyrolar is in his bloodstream immediately after having taken the fth pill? (c) Suppose Ali takes 3 grains of Thyrolar each day for one month. How much thyroid medication will be in his bloodstream right before he takes his 31 st pill? Right after? (d) After taking this medicine for many years, what was the amount of the drug in his body immediately after taking a pill? Historical note: Before one of his last ghts Muhammad Ali decided to up his dosage to 6 grains. In doing so he mimicked the symptoms of an overactive thyroid. The result in terms of the ght was dismal.

Make initial estimates to be sure that the answers you get are in the right ballpark. Suppose you borrow some money at an interest rate of \(6 \%\) compounded monthly. You begin paying back one month from today and make payments monthly. You pay back the entire debt after 180 payments of \(\$ 1000\) each. (This is a 15 -year mortgage.) How much money did you borrow?

For Problems 1 through 10, write the sum using summation notation. $$ 2^{3}+3^{3}+4^{3}+\cdots+19^{3} $$
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