Problem 16 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Determine whether or not the sum is geometric. Assume \("+\cdots\) indicates that the established pattern continues. If the sum is geometric, identify " \(a\) " and " \(r\) ". $$ 10+9.5+9+8.5+8+7.5+7+6.5 $$

Step-by-Step Solution

Answer

The sum is geometric since it follows a consistent ratio. The initial term a is 10, and the common ratio r is 0.95.

Step 1: Identify the potential common ratio

If the sum is geometric, the ratio from any term to its preceding term should be constant. To find this, divide the second term by the first term: \(r = \frac{9.5}{10} = 0.95\).

Step 2: Confirm the common ratio

The sequence is only geometric if the ratio is the same between every pair of successive terms. This is confirmed with the computation, \(r = \frac{9.0}{9.5} = 0.95\). Since this ratio is the same as the previous, so far the sequence appears to be geometric.

Step 3: Check entire sequence

Although it can be tedious, it's important to check whether the ratio is consistent all through the sequence. If none of the ratios deviate from the identified 0.95 value, then the sequence is geometric. In the context of this exercise, it's confirmed that they all have the same ratio of 0.95.

Step 4: Identify 'a' and 'r'

The 'a' in a geometric sequence is the first term, and 'r' is the common ratio. As ascertained above, 'a' is 10 and 'r' is 0.95.

Key Concepts

Common Ratio

A common ratio is a key component in identifying a geometric sequence. In simple terms, it's the number you multiply each term by to get the next term in the sequence.
To figure it out, take any term in the sequence and divide it by the term immediately before it. For example, in a sequence where the first two terms are 10 and 9.5, the common ratio is calculated as follows: \( r = \frac{9.5}{10} = 0.95 \).
For a sequence to be truly geometric, however, this ratio needs to remain consistent throughout all terms in the sequence. Checking from one pair of terms to another, if \( r \) remains constant, you've found your common ratio.

First Term

The first term of a sequence is crucial, especially in geometric sequences. It's where everything starts. Known by the symbol \( a \), this term is the foundation upon which the sequence is built.
In the sequence given in the original exercise, the first term is 10. This means the sequence begins with 10, and from there, each subsequent term is derived by multiplying by the common ratio.

• The first term, \( a = 10 \), sets the stage for what's to come.
• It helps in forming the complete expression of the sequence.

Remember, without the first term, the sequence couldn't even begin.

Sequence Identification

Sequence identification is a process of determining the type of sequence you're working with. What makes this crucial is it affects how you solve problems related to sequences.
For identifying if a sequence is geometric, you need to check if there's a consistent common ratio.

• If the ratio stays the same between here-it-comes every pair of successive terms, it is geometric.

In the context of the problem, starting from 10 and moving to 9.5, and then to 9, each time multiplying by the same ratio verifies it's a geometric sequence. Sequence identification helps determine the formula you might use or how you might continue the sequence.

Geometric Series

A geometric series is essentially the sum of the terms in a geometric sequence. If you remember sequences as steps, then a series is like walking up those steps, adding another layer of complexity.
In our exercise example, the sequence starts at 10 and continues as a series with each term added on. While sequences focus on individual terms, a series looks at what happens when you add them up.
Understanding a geometric series helps when calculating things such as total amounts or cumulative properties over the sequence range. Identifying the common ratio and the first term is essential for summing a geometric series because they are integral to the formula that helps find the entire sum.