Problem 44 Write each series as a sum of te... [FREE SOLUTION]

Chapter 13: Problem 44

Write each series as a sum of terms and then find the sum. $$ \sum_{i=2}^{6}(i+3)(i-4) $$

Short Answer

Expert verified

The sum is 13.

Step by step solution

Understand the Series

We need to write out the given series $\sum_{i=2}^{6}(i+3)(i-4)$ by finding all terms starting from $i = 2$ to $i = 6$.

Expand Each Term

Next, we expand each term in the series $ (i+3)(i-4) $. For example, $ (2+3)(2-4) = 5(-2) = -10 $. Repeat this for each value of $ i $ from 2 to 6.

Calculate Each Term

Calculate each term individually: \: $(2+3)(2-4) = -10$, $(3+3)(3-4) = -3$, $(4+3)(4-4) = 0$, $(5+3)(5-4) = 8$, $(6+3)(6-4) = 18$.

Sum All Terms

Add all the calculated terms together: $-10 + (-3) + 0 + 8 + 18 = 13$.

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.

series expansion

When we talk about series expansion, we're referring to the process of breaking down each term in a series to simplify calculations or to better understand the pattern within the series.
In the given exercise, the series is defined as $\sum_{i=2}^{6}(i+3)(i-4)$.

To expand each term, we take each value of $i$ from 2 to 6 and substitute them into $ (i+3)(i-4)$ and then perform the multiplication.
This method lets us see each term individually and helps in verifying each step:

For $i = 2$: $ (2+3)(2-4) = 5(-2) = -10$
For $i = 3$: $ (3+3)(3-4) = 6(-1) = -6$
For $i = 4$: $ (4+3)(4-4) = 7(0) = 0$
For $i = 5$: $ (5+3)(5-4) = 8(1) = 8$
For $i = 6$: $ (6+3)(6-4) = 9(2) = 18$

This expansion simplifies the process of identifying and calculating the value of each term in the series.

arithmetic series

An arithmetic series is a sequence of numbers in which the difference between consecutive terms remains constant.
Unlike the given exercise, which is not purely arithmetic, the concept of sums in arithmetic sequences can be crucial in understanding series calculations.
For example, if you have a simple arithmetic series like 2, 5, 8, 11,... with a common difference of 3, you can easily determine the sum using the formula:

$\text{Sum} = \frac{n}{2}(a + l)$

Where:

$n$ = number of terms
$a$ = first term
$l$ = last term

However, for more complex series like the one in the given exercise, breaking it down by expanding each term, as shown earlier, will likely be more practical.

sum of terms

After successfully expanding each term in the series, the next step is to find the sum of all these terms.
This involves adding each individual term together:

From our series: $ -10, -6, 0, 8, 18 $

We simply add these up:
$-10 + (-6) + 0 + 8 + 18 = 10$
This calculation concludes by providing the sum of the entire series.
Remember, properly expanding and calculating each term accurately is crucial for finding the correct overall sum.

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

Write each series as a sum of terms and then find the sum. $$ \sum_{i=2}^{6}(i+3)(i-4) $$

Short Answer

Step by step solution

Understand the Series

Expand Each Term

Calculate Each Term

Sum All Terms

Key Concepts

series expansion

arithmetic series

sum of terms

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Calculus

Decision Maths

Mechanics Maths

Theoretical and Mathematical Physics

Geometry

Study anywhere. Anytime. Across all devices.