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Chapter 11: Problem 17

Find the common ratio for each geometric sequence. $$75,15,3, \frac{3}{5}, \dots$$

Short Answer

Expert verified
The common ratio is \(0.2\).

Step by step solution

01

Title - Identify the First Term

Identify the first term of the geometric sequence, which is given as \(75\).
02

Title - Identify the Second Term

Identify the second term of the sequence, which is \(15\).
03

Title - Apply the Formula for Common Ratio

The common ratio \(r\) in a geometric sequence is found by dividing the second term by the first term using the formula \( r = \frac{a_2}{a_1} \), where \(a_1\) is the first term and \(a_2\) is the second term.
04

Title - Calculate the Common Ratio

Substitute the identified terms into the formula: \( r = \frac{15}{75} \).
05

Title - Simplify the Fraction

Simplify the fraction: \( r = \frac{15}{75} = \frac{1}{5} = 0.2 \). Thus, the common ratio is \(0.2\).

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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, the common ratio is a constant factor between consecutive terms. This makes it different from arithmetic sequences, where the difference between terms is constant. To find the common ratio, we simply divide any term by its preceding term. For example, in the given sequence 75, 15, 3, \(\frac{3}{5}\), we first identify the first two terms: 75 and 15. By dividing 15 by 75, we find the common ratio:
sequence identification
Sequence identification involves identifying all elements that make up a sequence. Often, the terms appear to follow a pattern, making them part of a geometric sequence. In our exercise, we start with the term 75. The next term is 15. This hints at multiplication by a constant. A straightforward inspection shows subsequent terms 3 and \( \frac{3}{5} \), maintaining a consistent pattern, which confirms that these values are part of a geometric sequence.
formula application
Once the sequence is identified, applying the right formula is crucial. For a geometric sequence, the common ratio is found using \( r = \frac{a_2}{a_1} \). This formula says to divide the second term by the first term. Substituting our terms 15 and 75, the calculation proceeds as follows: \( r = \frac{15}{75} \).
fraction simplification
Simplifying fractions is a key step in many math problems, ensuring the values are in their simplest form. In our case, simplifying \(r = \frac{15}{75} \) gives us; \( r = \frac{1}{5} = 0.2 \). This helps in easy understanding and further calculations. Simplification involves finding the greatest common divisor (GCD) of the numerator and the denominator, which, for 15 and 75, is 15. Therefore, \( r = \frac{15 / 15}{75 / 15} = \frac{1}{5} \).

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Most popular questions from this chapter

Form 1 of the binomial theorem can be proved using form 2 of the binomial theorem. The key step in that proof is showing that the coefficients inside Pascal's triangle are found by adding the two terms above. Prove this fact by showing that $$\left(\begin{array}{l}{n} \\\\{r}\end{array}\right)=\left(\begin{array}{c}{n-1} \\ {r-1}\end{array}\right)+\left(\begin{array}{c}{n-1} \\\\{r}\end{array}\right)$$

It is said that as a young child, the mathematician Karl F. Gauss \((1777-1855)\) was able to compute the sum \(1+2+3+\cdots+100\) very quickly in his head. Explain how Gauss might have done this and present a formula for the sum of the first \(n\) natural numbers. (Hint: \(1+99=100 .)\)

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