Problem 79 - Precalculus with Limits

Question

In Exercises 67 - 86, find the sum of the finite geometric sequence. \( \sum_{n=0}^{5}300\left(1.06\right)^n \)

Step-by-Step Solution

Answer

The sum of the given geometric sequence is approximately 2010.79.

Step 1: Understand the Formula

A geometric sequence is in the form: \(a, ar, ar^2, ar^3, ..., ar^{n-1}\) where \(r\) is the ratio between two consecutive terms. The sum \(\sum_{n=0}^{N}ar^n\) of a geometric sequence is given by the formula \(\frac{a(r^{N+1}-1)}{r-1}\) when \(r>1\).

Step 2: Identify the elements

In the given sequence, we can identify the first term \(a\) as 300, the ratio \(r\) as 1.06 and the number of terms \(N\) as 5.

Step 3: Substitute in the Sum Formula

Substitute \(a=300\), \(r=1.06\), and \(N=5\) into the sum formula: \(Sum = \frac{300(1.06^{5+1}-1)}{1.06-1}\).

Step 4: Calculate the Sum

Perform the calculations to find the sum of the sequence. Calculate \(1.06^{6}\), subtract one, multiply by 300, and divide the product by \(1.06-1\).

Key Concepts

Finite Series

A finite series is a sum of terms that stops at a particular number, unlike an infinite series that goes on forever.
In mathematics, we often encounter finite series when calculating the sum of geometric or arithmetic sequences.
In our exercise, we deal with a geometric sequence which is summed up to a certain number of terms.

• Each term in a finite series contributes to the sum.
• This particular sequence involves just six terms, making it finite.
• It's important to note that once you've summed all the specified terms, the series ends.

Understanding finite series is crucial as it helps in knowing the bounds within which you are summing up the terms. It simplifies real-world problems where the series naturally stops after a certain point.

Sum Formula

The sum formula for a geometric sequence helps us find the sum of all terms in the sequence without adding each one individually.
This is particularly useful for long sequences.
The sum formula for a geometric series is:\[S_N = \frac{a(r^{N+1} - 1)}{r - 1}\]Where:

• \(S_N\) is the sum of the series,
• \(a\) is the first term,
• \(r\) is the common ratio,
• \(N\) is the number of terms minus one.

This mathematical formula takes the hassle out of summing each term individually, especially when the number of terms is large.
By substituting the values of the first term, common ratio, and number of terms into the formula, the sum of the entire sequence can be calculated swiftly.

Geometric Progression

A geometric progression, also known as a geometric sequence, is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.
Think of it like this:

• Start with the first term \(a\).
• Multiply it by the common ratio \(r\) to get the next term.
• Repeat this process to generate more terms.

For example, in the sequence given in the exercise, you start at 300 and multiply by 1.06 to get the next term. This replicates for each subsequent term until the required number of terms is reached.
Understanding geometric progression is key to solving many problems, especially involving exponential growth or decay in various fields such as finance, population studies, and computer science.

Common Ratio

The common ratio in a geometric progression is the factor by which each term is multiplied to get to the next term.
It is a constant that remains the same throughout the entire sequence.In the context of our exercise:

• The common ratio \(r\) is 1.06.
• It signifies a 6% increase in successive terms since multiplying by 1.06 incrementally increases each term by 6% over the previous one.
• If \(r\) is greater than 1, like in our example, the sequence is increasing.
• If \(r\) is less than 1 but greater than 0, the sequence decreases.

The common ratio is a fundamental element of the geometric sequence, as it dictates how each new term is derived from the one before it.
It provides a systematic and predictable way to generate new terms in the sequence.