Problem 68 Evaluate each sum using a formul... [FREE SOLUTION]

Chapter 15: Problem 68

Evaluate each sum using a formula for $S_{n}$. $$\sum_{i=1}^{215}(i-10)$$

Short Answer

Expert verified

We are given the arithmetic series $\sum_{i=1}^{215}(i-10)$ with $n = 215$, $a_1 = -9$, and $a_n = 205$. Using the formula for the sum of an arithmetic series, $S_n = n\cdot \frac{a_1 + a_n}{2}$, we find the sum: $S_n = 215 \cdot \frac{-9 + 205}{2} = 215 \cdot 98 = 21070$.

Step by step solution

Find the first and last terms of the series

To find the first term $a_1$, plug in the value of $i = 1$ into the expression $(i - 10)$: $$a_1 = 1 - 10 = -9$$ To find the last term $a_n$, plug in the value of $i = 215$ into the expression $(i - 10)$: $$a_n = 215 - 10 = 205$$

Plug in the values to the formula and solve for the sum $S_n$

We have found $n = 215$, $a_1 = -9$, and $a_n = 205$. Plugging these values into the formula for the sum of an arithmetic series, we have: $$S_n = 215 \cdot \frac{-9 + 205}{2}$$

Calculate the result of the sum $S_n$

Now, evaluate the expression to find the sum $S_n$: $$S_n = 215 \cdot \frac{196}{2} = 215 \cdot 98 = 21070$$ The sum of the given series is $21070$.

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Sum Formula

Understanding the "Sum Formula" is essential when dealing with arithmetic series. An arithmetic series is the sum of the terms of an arithmetic sequence. To efficiently compute the sum, you can use the formula:
\[ S_n = \frac{n}{2} \times (a_1 + a_n) \]Here, $S_n$ represents the sum of the series, $n$ is the number of terms, $a_1$ is the first term, and $a_n$ is the last term in the sequence.
The formula leverages the property of evenly spaced numbers in an arithmetic sequence. It combines the first and last terms and averages them by dividing by two. Then, you multiply by the total number of terms to obtain the sum.

This saves time compared to individually adding each term, especially with a large number of terms.
Understanding this formula helps you tackle various problems involving arithmetic sequences efficiently.

Arithmetic Sequence

When talking about an "Arithmetic Sequence," it's important to understand that it's a list of numbers where each term increases by a constant amount, called the common difference $d$. The general form of an arithmetic sequence is:
\[ a_n = a_1 + (n - 1) \, d \]where $a_n$ is the $n$-th term, $a_1$ is the first term, and $d$ is the common difference between the terms.
In the given exercise, the expression $(i - 10)$ defines the terms of the sequence:

The first term $a_1$ is found by substituting $i = 1$, giving us $a_1 = -9$.
The last term $a_{215}$ is calculated by substituting $i = 215$, which gives us $a_{215} = 205$.

Understanding these sequences is crucial for identifying patterns and determining the sum of the sequence efficiently.

Sequence Evaluation

"Sequence Evaluation" involves gathering information about a sequence to reach a conclusion. With arithmetic sequences, you often aim to find the sum or specific terms within the sequence.
In our example, by evaluating the sum through an arithmetic sequence, we:

Identified the first term $a_1 = -9$ and the last term $a_{215} = 205$ using the given expression $(i - 10)$.
Determined the number of terms $n = 215$ in the sequence.

After calculating these, we applied the sum formula to determine the total of the series.
Sequence evaluation helps you look past individual numbers to see the broader mathematical relationships and enables you to apply formulas systematically. Critically evaluating sequences ensures accuracy in solving complex problems.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

Evaluate each sum using a formula for \(S_{n}\). $$\sum_{i=1}^{215}(i-10)$$

Short Answer

Step by step solution

Find the first and last terms of the series

Plug in the values to the formula and solve for the sum \(S_n\)

Calculate the result of the sum \(S_n\)

Key Concepts

Sum Formula

Arithmetic Sequence

Sequence Evaluation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Statistics

Applied Mathematics

Mechanics Maths

Calculus

Logic and Functions

Study anywhere. Anytime. Across all devices.