Problem 25 - Contemporary Precalculus

Question

In Exercises \(23-30,\) show that the given sequence is geometric and find the common ratio. $$\left\\{5^{n+2}\right\\}$$

Step-by-Step Solution

Answer

Question: Show that the sequence given by the formula \(5^{n+2}\) is a geometric sequence and find the common ratio. Answer: The sequence is a geometric sequence with a common ratio of 5.

Step 1: Identifying consecutive terms of the sequence

To analyze a geometric sequence, we need to find the consecutive terms of the sequence, which are given by: Term \(n\): \(a_n = 5^{n+2}\) Term \(n+1\): \(a_{n+1} = 5^{n+3}\)

Step 2: Finding the ratio between consecutive terms

To check if a sequence is geometric, we need to find the ratio between consecutive terms and check if it is constant. We can do this by dividing the term \(n+1\) by term \(n\), as follows: $$\frac{a_{n+1}}{a_n} = \frac{5^{n+3}}{5^{n+2}}$$

Step 3: Simplifying the expression

Now, we need to simplify the expression by using the properties of exponents: $$\frac{5^{n+3}}{5^{n+2}} = 5^{(n+3)-(n+2)}$$ $$= 5^{n+3-n-2} = 5^1$$

Step 4: Conclusion

Since the ratio between consecutive terms is \(5^1 = 5\), which is constant, we can conclude that the given sequence is a geometric sequence. And, the common ratio is 5.

Key Concepts

Common Ratio

In geometric sequences, the common ratio is a crucial component. It determines the pattern used to establish each term in the sequence from the previous one. Specifically, to find a common ratio, you divide any term by the preceding term. For example, in a sequence of numbers such as 2, 4, 8, and 16, dividing any term by the previous one (4/2, 8/4, 16/8) always gives the same result - in this case, the common ratio is 2.

This means that each term is twice as large as the previous term. The property of having a constant common ratio defines a geometric sequence. A sequence that doesn't exhibit this pattern, where you cannot find such a consistent ratio, is not classified as geometric. In the textbook example \(5^{n+2}\), the common ratio is found by dividing a term by its preceding term, resulting in the constant value of 5, hence confirming its geometric nature.

Exponents

Exponents play a key role in geometric sequences. They represent how many times to multiply a number by itself. For instance, \(5^3\) means \(5 \times 5 \times 5\). When dealing with geometric sequences that involve exponents, understanding how to manipulate them is essential.

In our exercise, to get from \(5^{n+2}\) to the next term \(5^{n+3}\), you simply multiply by 5, or add 1 to the exponent \(n+2\) to obtain \(n+3\). Simplifying the expression \(\frac{5^{n+3}}{5^{n+2}}\) using exponent rules, you subtract the exponents because of the division (which gives you \(5^{(n+3)-(n+2)} = 5^1\)). This technique is a fundamental aspect of solving geometric sequences with exponent terms.

Consecutive Terms

The concept of consecutive terms is utilized to determine if a sequence is geometric. Consecutive terms in a sequence are simply terms that follow one after the other. In a sequence like 3, 6, 12, '6' and '12' are consecutive terms, just as '3' and '6' are.

Geometric sequences are determined by the consistent ratio between these consecutive terms. To prove this pattern, you only need to take any two consecutive terms and divide them. If the ratio is the same across the entire sequence, it is geometric. This constant relationship between consecutive terms is what creates the predictability and structure within a geometric sequence. For example, in our exercise, the term after \(5^{n+2}\) would logically be \(5^{(n+1)+2}\), demonstrating the sequence's predictable pattern of consecutive terms defined by the common ratio.