Problem 24 - Algebra 2 and Trigonometry

Question

In \(19-24 :\) a. Write each arithmetic series as the sum of terms. b. Find the sum. $$ \sum_{n=1}^{10}(-1+2 n) $$

Step-by-Step Solution

Answer

The sum of the series is 100.

Step 1: Understand the Arithmetic Expression

The given series is given in the form of \( \sum_{n=1}^{10}(-1+2n) \). This means we need to substitute \( n \) with each integer from 1 to 10 in the expression \( -1 + 2n \) and then sum the resulting terms.

Step 2: Generate the Terms of the Series

Substitute \( n = 1, 2, ..., 10 \) into the expression \(-1 + 2n\) to get the terms:- For \( n = 1 \): \( -1 + 2(1) = 1 \)- For \( n = 2 \): \( -1 + 2(2) = 3 \)- For \( n = 3 \): \( -1 + 2(3) = 5 \)- For \( n = 4 \): \( -1 + 2(4) = 7 \)- For \( n = 5 \): \( -1 + 2(5) = 9 \)- For \( n = 6 \): \( -1 + 2(6) = 11 \)- For \( n = 7 \): \( -1 + 2(7) = 13 \)- For \( n = 8 \): \( -1 + 2(8) = 15 \)- For \( n = 9 \): \( -1 + 2(9) = 17 \)- For \( n = 10 \): \( -1 + 2(10) = 19 \)Thus, the series of terms is: \( 1, 3, 5, 7, 9, 11, 13, 15, 17, 19 \).

Step 3: Calculate the Sum of the Series

To find the sum of the series, add all the terms together: \[1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19\]Adding these step-by-step:- \( 1 + 3 = 4 \)- \( 4 + 5 = 9 \)- \( 9 + 7 = 16 \)- \( 16 + 9 = 25 \)- \( 25 + 11 = 36 \)- \( 36 + 13 = 49 \)- \( 49 + 15 = 64 \)- \( 64 + 17 = 81 \)- \( 81 + 19 = 100 \)So, the sum of the arithmetic series is 100.

Key Concepts

Sum of Series

An arithmetic series is a sum of terms where each term increases by a constant, known as the difference. To find the sum of an arithmetic series, we add each term together.
In our exercise, the series is generated using the expression \(-1 + 2n\), with \(n\) from 1 to 10.
We calculated the sum by first creating the individual terms and then adding them:

• Start with the first term \(1\)
• Continue adding each subsequent term \(3, 5, 7, \dots\)
• Until you reach the last term \(19\)

You could also use a formula for the sum of an arithmetic series if the terms follow a clear pattern:\[S = \frac{n}{2} (a + l)\]where:- \(S\) is the sum of the series- \(n\) is the number of terms- \(a\) is the first term- \(l\) is the last termUsing our given series:- First term \(a = 1\)- Last term \(l = 19\)- Number of terms \(n = 10\)
Insert these into the formula:\[S = \frac{10}{2} (1 + 19) = 5 \times 20 = 100\]This matches our step-by-step solution where we manually added each term.

Substitution Method

The substitution method is a simple yet powerful tool used to generate terms in a sequence or series.
You take an expression involving a variable and substitute specific values for that variable to generate terms.
In our arithmetic series exercise, we have the expression \(-1 + 2n\) and substitute \(n\) with integers from 1 to 10.

• Begin with \(n = 1\), substitute to get \(-1 + 2(1) = 1\)
• Continue with \(n = 2\), substitute to get \(-1 + 2(2) = 3\)
• Repeat this until \(n = 10\), giving \(-1 + 2(10) = 19\)

This method ensures that each term is calculated precisely by following the defined equation or rule,
producing the sequence \(1, 3, 5, 7, \dots, 19\).
It's helpful not only for arithmetic series but also for identifying patterns and generating specific values within a sequence in various areas of mathematics.

Term Generation

Generating terms in an arithmetic sequence from an expression involves creating a list of numbers that share a common difference. This is fundamental to understanding how series work.
In our exercise:- The common difference is determined by the expression \(-1 + 2n\).
Every time \(n\) increases by 1, the result increases by 2, making the common difference \(d = 2\).- Start with the value of \(n = 1\), producing the first term, then repeatedly increase \(n\) by 1 to get following terms.The resulting sequence from \(n=1\) to \(n=10\) becomes \(1, 3, 5, 7, \dots, 19\), consistent with arithmetic principles.
To quickly check if your generated sequence is correct, verify that the difference between each consecutive pair of terms is consistent:

• \(3 - 1 = 2\)
• \(5 - 3 = 2\)
• And so forth, until \(19 - 17 = 2\)

Each step verifies the arithmetic measure that confirms constant growth, making it a critical tool in both learning and applying arithmetic sequences.