Problem 54 - Algebra and Trigonometry

Question

express each sum using summation notation. Use 1 as the lower limit of summation and i for the index of summation. $$ a+a r+a r^{2}+\dots+a r^{n-1} $$

Step-by-Step Solution

Answer

The sequence \(a+a r+a r^{2}+\dots +a r^{n-1}\) can be expressed as \(\sum_{i=1}^n a r^{i-1}\) using the summation notation.

Step 1: Identify the Sequence

First, we recognize that the sequence given is a geometric sequence. It can be written as the series \(a+a r+a r^{2}+\dots +a r^{n-1}\). Each term is the product of the previous term and a fixed number \(r\), known as the common ratio.

Step 2: Identify the General Term

In this sequence, the general term is given by \(a r^{i-1}\), where \(i\) starts from 1 up to \(n\). It's very important to recognize that the base of \(r\) increases by one with each term.

Step 3: Apply the Summation Notation

Now we introduce the sigma notation (\(\Sigma\)), which means sum. Summation notation allows us to write this series in a more compact form. We let \(i\) be our index of summation, making sure it varies from 1 to \(n\), the number of terms in our series, and summarize the sequence within the sigma notation as \(\sum_{i=1}^n a r^{i-1}\).

Key Concepts

Geometric Sequence

A geometric sequence is a type of sequence where each term after the first is obtained by multiplying the previous term by a fixed, non-zero number called the common ratio. This type of sequence is very different from an arithmetic sequence, where you add a constant number to get from one term to the next. In the exercise given, we look at the sequence:

• Starts with an initial term, often denoted as \(a\).
• Each subsequent term is obtained by multiplying the preceding term by a certain number \(r\), called the common ratio.

So, if the initial term is \(a\) and the common ratio is \(r\), the sequence will look like \(a, ar, ar^2, ar^3, \ldots\). Note how the powers of \(r\) increase with each term, showcasing the nature of a geometric sequence. This setup helps in quickly recognizing patterns in sequences, making it easier to perform calculations and understanding their behavior.

Common Ratio

The common ratio, denoted as \(r\) in a geometric sequence, is a critical component that defines the progression of terms. It signifies how the sequence grows or shrinks with each subsequent term. Here's how to identify and understand the significance of the common ratio:

• To find the common ratio, divide any term in the sequence by its preceding term.
• Mathematically, if the first term is \(a\), the second is \(ar\), then the common ratio \(r\) is computed as \( \frac{ar}{a} = r\).
• The value of \(r\) determines the nature of the sequence:
  • If \(|r| > 1\), the sequence is said to grow exponentially.
  • If \(0 < |r| < 1\), the sequence will shrink and approach zero.
  • If \(r = 1\), the sequence is constant, as each term is the same as the initial term.

Appreciating the role of the common ratio helps in predicting the behavior of the sequence, which is crucial for solving problems and applying summation notation effectively.

General Term

In any sequence, understanding the "general term" or the formula for each term based on its position is essential. This concept allows us to not only find any specific term in a sequence easily but also sum up a sequence using summation notation. In the context of our geometric sequence:\[ a, ar, ar^2, \ldots, ar^{n-1} \]

• The general term for any \(i^{th}\) term in a geometric sequence can be expressed as \(ar^{i-1}\).
• \(a\) is the first term, \(r\) is the common ratio, and \(i\) is the position of the term in the sequence starting from 1.
• This formula allows us to directly compute any term's value without listing all the terms before it.

By mastering the general term, you're able to apply \( \Sigma \) notation more seamlessly, because you know the exact formula used in each term of the sequence. This simplification is particularly useful when working with lengthy sequences and sums.