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Chapter 9: Problem 40

Find the indicated term of the geometric sequence. $$\text { 7th term: } 3,36,432, . . .$$

Short Answer

Expert verified
The 7th term of the geometric sequence is 3,796,594.

Step by step solution

01

Determine the common ratio

The common ratio, r, is determined by dividing the second term by the first term, thus, \( r = \frac{36}{3} = 12\) .
02

Apply the formula

The formula for the nth term of a geometric sequence is \(a_n=a_1 \times r^{(n-1)}\). Substituting the given values, we get \(a_7 = 3 \times 12^{(7-1)}\) . This calculation gives the value of the 7th term.
03

Calculate the 7th term

Calculate the 7th term by simplifying the expression from step 2, \(a_7 = 3 \times 12^{6}\), which evaluates to 3,796,594.

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Key Concepts

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

Common Ratio
In a geometric sequence, every term is generated by multiplying the previous term by a fixed number, known as the "common ratio." This concept is useful to identify how the sequence progresses. To find this ratio, simply divide any term in the sequence by the one that precedes it. For example, when given the sequence 3, 36, 432, you can find the common ratio by dividing the second term (36) by the first term (3). This gives the result, 12. This means each term can be calculated by multiplying the previous term by 12.
  • Understanding the common ratio is crucial because it determines the pattern of growth in the sequence.
  • If the ratio is greater than 1, the sequence grows; if it's between 0 and 1, it shrinks.
  • If the ratio is 1 or less than 0, the series becomes constant or oscillates back and forth.
Knowing how to find and interpret the common ratio helps understand broader patterns, either in finances, population growth, or even algorithm performance.
Nth Term Formula
The Nth term formula in a geometric sequence allows you to find any term without listing all previous terms. It鈥檚 like having a shortcut! This formula is given by \(a_n = a_1 \times r^{(n-1)}\).
  • Here, \(a_1\) is the first term of the sequence.
  • \(r\) stands for the common ratio.
  • \(n\) is the position of the term you want to find.
  • \(a_n\) is the value of the term at position \(n\).
For example, if you want to find the 7th term of the sequence starting with 3, with a common ratio of 12, simply substitute these values into the formula: \(a_7 = 3 \times 12^{6}\),which helps you easily find the desired term. This formula is a powerful tool that allows you to quickly determine the nature of a sequence without exhaustive computations.
Sequence Calculation
Sequence calculation involves using formulas and arithmetic to obtain specific terms from a given sequence. In a geometric sequence, these calculations revolve around properly applying the common ratio and Nth term formula. Here鈥檚 how you practically approach it:
  • Identify the first term and the common ratio.
  • Use the Nth term formula to calculate the desired term.
For instance, to find the 7th term of the sequence 3, 36, 432:
  • First, identify \(a_1 = 3\) and \(r = 12\).
  • Substitute these into the formula \(a_n = a_1 \times r^{(n-1)}\) to get \(3 \times 12^{6}\).
  • Simplify \(3 \times 12^{6} = 3,796,594\), which is the 7th term.
Breaking it down this way highlights how systematic sequence calculation provides clear pathways to solving sequence problems efficiently.

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Most popular questions from this chapter

Use the Binomial Theorem to approximate the quantity accurate to three decimal places. For example, in Exercise \(79,\) use the expansion \(\begin{aligned}(1.02)^{8} &=(1+0.02)^{8} \\ &=1+8(0.02)+28(0.02)^{2}+\cdot \cdot \cdot+(0.02)^{8}\end{aligned}\), $$(2.005)^{10}$$

Without calculating, determine whether the value of \(_{n} P_{r}\) is greater than the value of \(_{n} C_{r}\) for the values of \(n\) and \(r\) given in the table. Complete the table using yes (Y) or no (N). Is the value of \(_{n} P_{r}\) always greater than the value of \(_{n} C_{r} ?\) Explain.

Simplify the difference quotient, using the Binomial Theorem if necessary.\(\frac{f(x+h)-f(x)}{h}\). $$f(x)=\sqrt{x}$$

True or False? Determine whether the statement is true or false. Justify your answer. Rolling a number less than 3 on a normal six-sided die has a probability of \(\frac{1}{3} .\) The complement of this event is to roll a number greater than \(3,\) and its probability is \(\frac{1}{2}.\)

Random Number Generator A random number generator on a computer selects two integers from 1 through \(40 .\) What is the probability that (a) both numbers are even, (b) one number is even and one number is odd, (c) both numbers are less than \(30,\) and (d) the same number is selected twice?
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