91Ó°ÊÓ

Summation notation is a compact way to express the addition of a sequence of terms. It uses the Greek letter sigma (\( \Sigma \)) to represent the sum. In the given exercise, the summation notation is \( \sum_{k=1}^{6} 100 \left( \frac{1}{2} \right)^{k} \). This tells us to sum the terms \( 100 \left( \frac{1}{2} \right)^{k} \), starting with \( k = 1 \) and ending with \( k = 6 \).

• The expression under the sigma symbol is the general term that will be repeated.
• The numbers below and above the sigma (1 and 6 in this case) denote the first and last values of \( k \).
• By expanding this notation, we perform the calculation: \( 100 \times \left(\frac{1}{2}\right) + 100 \times \left(\frac{1}{4}\right) + ... + 100 \times \left(\frac{1}{64}\right) \)

Summation notation is helpful in simplifying expressions that involve many terms by providing a shorthand that is easy to understand and use.

Series expansion involves taking an expression in summation notation and writing it out in full. In the example provided, the summation \( \sum_{k=1}^{6} 100 \left( \frac{1}{2} \right)^{k} \) is expanded into a series by substituting values for \( k \) from 1 to 6. This process includes the following steps:

• Replace \( k \) with \( 1 \) to get the first term: \( 100 \times \frac{1}{2} \)
• Replace \( k \) with \( 2 \) to get the second term: \( 100 \times \frac{1}{4} \)
• Continue this pattern until \( k \) equals 6.

The result is a series of terms: \[ 100 \left( \frac{1}{2} \right)^{1} + 100 \left( \frac{1}{2} \right)^{2} + 100 \left( \frac{1}{2} \right)^{3} + 100 \left( \frac{1}{2} \right)^{4} + 100 \left( \frac{1}{2} \right)^{5} + 100 \left( \frac{1}{2} \right)^{6} \]Writing expressions as a series makes them easier to understand and calculate, as it shows every term involved in the sum.

A sequence is an ordered list of numbers following a particular pattern, while a series is the sum of the terms of a sequence. In this context, the terms obtained from the geometric sequence \( 100, 50, 25, 12.5, 6.25, 3.125, 1.5625 \) build the series we are working with.

• A geometric sequence is one where each term is a constant multiple of the previous term. Here, each term is multiplied by \( \frac{1}{2} \) from the previous term.
• We sum these terms to form the series: \( 50 + 25 + 12.5 + 6.25 + 3.125 + 1.5625 \).

Studying sequences and series allows us to understand patterns in numbers and to calculate sums efficiently using mathematical rules and formulas. The sequence progression showcases how each component contributes to the total sum in a measurable way.

Mathematical expressions are combinations of numbers, symbols, and operation signs that represent a specific value or set of values. In the exercise \( 100+\sum_{k=1}^{6} 100\left(\frac{1}{2}\right)^{k} \), we work with a mathematical expression that includes:

• The constant term \( 100 \)
• A summation part \( \sum_{k=1}^{6} 100\left(\frac{1}{2}\right)^{k} \)

Mathematical expressions help us formulate problems and solutions in a structured way. They encapsulate complex ideas in a precise manner, allowing us to manipulate and evaluate them to find an answer. In the end, calculating the sum entails combining each evaluated part of the series and the constant, leading to the final value of \( 198.4375 \).