Problem 46 - Algebra and Trigonometry with Analytic Geometry

Question

$\sum_{k=137}^{428} 2.1 $$

Step-by-Step Solution

Answer

The sum is 613.2.

Step 1: Understand the Summation Expression

The summation notation \( \sum_{k=137}^{428} 2.1 \) means that we are adding 2.1 repeatedly for each integer value of \( k \) starting from 137 to 428.

Step 2: Determine the Number of Terms

To find how many times we need to add 2.1, we calculate the number of terms in the series, which is \( 428 - 137 + 1 = 292 \). This tells us that there are 292 terms.

Step 3: Find the Total Sum

To find the sum, multiply the value being summed (2.1) by the number of terms: \( 2.1 \times 292 \). Calculate this to find the total sum.

Step 4: Perform the Calculation

Perform the multiplication: \( 2.1 \times 292 = 613.2 \). This is the total sum of the series.

Key Concepts

Series

In mathematics, a series is essentially the sum of terms of a sequence. When we talk about summing up numbers in a series, we are referring to the continuous addition of numbers that follow a particular order. In simpler terms, imagine writing a sequence of numbers and then figuring out what you get when you add them all together.
For example, if you have the numbers 1, 2, and 3, the series would be the result of adding 1 + 2 + 3, which equals 6.
Series can have various forms:

• Finite Series: A series with a definite number of terms.
• Infinite Series: A series that goes on endlessly without a stopping point.

Series can also be classified based on the type of numbers or patterns they form, such as arithmetic or geometric. The original exercise showcases a finite series because it has a clear beginning point (137) and an endpoint (428). By understanding how the series is structured, students can more easily calculate the sums involved.

Arithmetic Sequence

An arithmetic sequence is a sequence of numbers in which the difference between any two consecutive terms is the same. This difference is referred to as the "common difference."
For example, consider the sequence 2, 4, 6, 8. Here, each term increases by 2, which is the common difference.
Arithmetic sequences can be found in various places, such as:

• Patterns in number systems.
• Daily life schedules (e.g., train timings).
• Mathematical problems as shown in the original exercise.

The importance of understanding an arithmetic sequence lies in its predictability, making it easy to find future numbers in the sequence. In the context of the original exercise, although the number 2.1 is not changing, the arithmetic sequence here lies in the positions of the terms (137 to 428). This consistency simplifies the calculation of the series.

Summation Notation

Summation notation is a mathematical convention used to represent the sum of a series. It is represented by the Greek letter sigma (\( \sum \)), and is a convenient way to write long sums of numbers without listing all the terms individually.
Here's what the components of summation notation mean:

• The lower limit, which is the number at the bottom of the sigma, indicates where the sum starts (e.g., \( k = 137 \)).
• The upper limit at the top tells you where the sum ends (e.g., \( k = 428 \)).
• The expression next to the sigma (\( 2.1 \) in this case) denotes the value to be repeatedly added, or the function to be applied to each term.

Understanding Summation:
Summation notation is particularly helpful in mathematics, as it offers a streamlined format for managing large sums.They eliminate the need of writing each number and plus sign and provide clarity, especially when dealing with complex sequences.
The exercise uses summation notation effectively to show that we take the constant term 2.1 and add it through each integer value of \( k \) from 137 to 428. This is a great example of how summation notation simplifies often cumbersome tasks.