Problem 52 - College Algebra

Question

Express each sum using summation notation. Use I as the lower limit of summation and i for the index of summation. $$-\frac{1}{9}+\frac{2}{9^{2}}+\frac{3}{9^{3}}+\dots+\frac{n}{9^{n}}$$

Step-by-Step Solution

Answer

Expressed as summation notation, the series becomes \( \sum_{i=1}^{n} \frac{i}{9^{i}} \).

Step 1: Identify the type of sequence

This is a geometric sequence where each term is multiplied by \( \frac{1}{9} \) to get the next term. The terms given are \( -\frac{1}{9} \), \( \frac{2}{9^{2}} \), \( \frac{3}{9^{3}} \), and so on. Each term \( t_i \) is given by the formula \( \frac{i}{9^{i}} \).

Step 2: Express in summation notation

In summation notation, the series can be written as the sum from \( i=1 \) to \( n \) of the expression \( \frac{i}{9^{i}} \). That is, it can be expressed as \( \sum_{i=1}^{n} \frac{i}{9^{i}} \).

Key Concepts

Geometric Sequence

A geometric sequence is a list of numbers where each number, after the first, is found by multiplying the previous one by a fixed, non-zero number called the common ratio. For example, in the sequence \(-\frac{1}{9}, \frac{2}{9^2}, \frac{3}{9^3}, \ldots\), each term is part of a geometric sequence where the common ratio is \(\frac{1}{9}\).

• Start with any number as your first term.
• Multiply each following term by the common ratio.

The regular pattern and predictability make geometric sequences easy to understand and work with, especially as they often simplify complex problems into manageable steps.

Series

A series can be understood as the sum of the terms in a sequence. In the context of sequences, specifically, we often deal with infinite or finite series. When we have a geometric sequence, each term is added together to form a geometric series.

• Add all terms in the sequence.
• Can be finite (with an upper limit) or infinite.

For the provided problem, the sequence \(-\frac{1}{9}, \frac{2}{9^2}, \frac{3}{9^3}, \ldots\) is summed up from the first to the nth term, forming a finite series, expressed using summation notation.

Index of Summation

The index of summation is crucial in working with series in algebra and calculus. It serves as a placeholder or variable within the summation notation. Typically represented by a letter like \(i\), it tells you where to start (the lower limit) and where to end (the upper limit). In our example, the index of summation \(i\) runs from \(1\) to \(n\).

• Identifies each term position in the series.
• Informs the computation order.

Using the index of summation allows mathematicians to handle large series by simplifying and compacting the information into a single expression.

Sequence Formula

A sequence formula is a mathematical expression that allows the calculation of any term in the sequence from its position number. For geometric sequences, the formula is comprised of a common ratio raised to the power of its order in the sequence.

• Formula Format: \(a_i = a \cdot r^{i-1}\), where \(a\) is the first term and \(r\) is the common ratio.
• Specifically, for our sequence, it becomes \(\frac{i}{9^i}\).

By using the sequence formula, getting to any term in the sequence becomes quick and efficient, as opposed to manually calculating each one successively.