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Chapter 9: Problem 69

Use summation notation to write the sum. $$10+30+90+\dots+7290$$

Short Answer

Expert verified
The summation notation for the sum of the sequence is \(\displaystyle\sum_{n=1}^{N} 10 * 3^{n-1}\)

Step by step solution

01

Identifying the pattern

This is a sequence where every term after the first is found by multiplying the previous term by 3. In other words, to go from one term to the next, multiply by 3. For example, \(\frac{30}{10} = 3 \), \(\frac{90}{30} = 3 \), and so on. This means that we are dealing with a geometric sequence.
02

Formulating the general term

A general term of a geometric sequence can be given by \(a * r^{n-1}\), where a is the first term, r is the common ration and n is the term number. In our case, a is 10, r is 3 and n is the term number. So the general term \(t_n\) of our sequence is \(10 * 3^{n-1}\). This formula will generate the nth term of our sequence.
03

Convert to summation notation

Now that we have a general term, we can express the whole sequence in summation notation. Since the series starts at n=1 and assuming it ends at n=N, the summation notation will be \(\displaystyle\sum_{n=1}^{N} 10 * 3^{n-1}\). The limits of the sum (1 to N) may be replaced with the actual start and end terms of the sequence when N is known.

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Key Concepts

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

Understanding Geometric Sequences
A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a constant value called the common ratio. This means that each term is a constant multiple of the term immediately before it.
A geometric sequence has several key characteristics which make it distinct:
  • Each term is obtained by multiplying the preceding term by a fixed, non-zero number known as the common ratio.
  • The pattern of multiplication remains consistent throughout the entire sequence.
  • Geometric sequences can either converge or diverge based on the magnitude of the common ratio.
Recognizing a geometric pattern is crucial because it helps in predicting future terms and solving related mathematical problems. In the given sequence, 10, 30, 90, ..., 7290, it is evident that each term is being multiplied by 3 to get the next term.
What is a Common Ratio?
The common ratio in a geometric sequence is the constant factor that each term is multiplied by to get the next term. It plays a fundamental role in defining the nature of the sequence. To find the common ratio, divide any term by its preceding term.
For example, in the sequence 10, 30, 90, ..., the common ratio is calculated as follows:
  • From 10 to 30: \(\frac{30}{10} = 3\)
  • From 30 to 90: \(\frac{90}{30} = 3\)
The common ratio, in this case, is 3. This means every term is 3 times the previous term. It is important because it defines how quickly or slowly the sequence grows or shrinks depending on whether the common ratio is greater than, equal to, or less than one.
Defining the General Term
The general term of a geometric sequence provides a formula to find any term in the sequence without having to list all previous terms. It's expressed using the formula: \[ t_n = a \cdot r^{n-1} \] where:
  • \(t_n\) is the nth term you want to find.
  • \(a\) is the first term of the sequence.
  • \(r\) is the common ratio.
  • \(n\) is the term number.
In the example sequence, where the first term \(a\) is 10 and the common ratio \(r\) is 3, the general term is:\[ t_n = 10 \cdot 3^{n-1} \] This formula allows us to plug in any value of \(n\) to determine the corresponding term in the sequence. Using the general term formula is a powerful tool for understanding and working with geometric sequences, as it provides a concise mathematical representation of the entire sequence.

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Most popular questions from this chapter

Finding the Probability of a Complement You are given the probability that an event will happen. Find the probability that the event will not happen. $$P(E)=0.87$$

Finding the Probability of a Complement You are given the probability that an event will not happen. Find the probability that the event will happen. $$P\left(E^{\prime}\right)=0.23$$

Consider \(n\) independent trials of an experiment in which each trial has two possible outcomes: "success" or "failure." The probability of a success on each trial is \(p,\) and the probability of a failure is \(q=1-p .\) In this context, the term \(_{n} C_{k} p^{k} q^{n-k}\) in the expansion of \((p+q)^{n}\) gives the probability of \(k\) successes in the \(n\) trials of the experiment.The probability of a baseball player getting a hit during any given time at bat is \(\frac{1}{4} .\) To find the probability that the player gets three hits during the next 10 times at bat, evaluate the term $$_{10} C_{3}\left(\frac{1}{4}\right)^{3}\left(\frac{3}{4}\right)^{7}$$, in the expansion of \(\left(\frac{1}{4}+\frac{3}{4}\right)^{10}\).

Prove the property for all integers \(r\) and \(n\) where \(0 \leq r \leq n\).The sum of the numbers in the \(n\) th row of Pascal's Triangle is \(2^{n}\).

The expansions of \((x+y)^{4},(x+y)^{5},\) and \((x+y)^{6}\) are as follows.$$\begin{aligned} (x+y)^{4}=& 1 x^{4}+4 x^{3} y+6 x^{2} y^{2}+4 x y^{3}+1 y^{4} \\\\(x+y)^{5}=& 1 x^{5}+5 x^{4} y+10 x^{3} y^{2}+10 x^{2} y^{3} \\\\(x+y)^{6}=& 1 x^{6}+6 x^{5} y+15 x^{4}+1 y^{5} \\\& \quad+6 x y^{5}+1 y^{6}\end{aligned}$$,(a) Explain how the exponent of a binomial is related to the number of terms in its expansion. (b) How many terms are in the expansion of \((x+y)^{n} ?\)
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