https://file.aiquickdraw.com/mathbot/file/1726798708482l10mv6i3.jpg

The given image shows a summation notation:

\[
\sum_{i=1}^{10} (3i^3 - 5)
\]

This represents the sum of the expression \( 3i^3 - 5 \) as \( i \) runs from 1 to 10. Let’s break this down step-by-step:

1. For \( i = 1 \), the expression is: 
   \[
   3(1^3) - 5 = 3(1) - 5 = 3 - 5 = -2
   \]

2. For \( i = 2 \), the expression is: 
   \[
   3(2^3) - 5 = 3(8) - 5 = 24 - 5 = 19
   \]

3. For \( i = 3 \), the expression is: 
   \[
   3(3^3) - 5 = 3(27) - 5 = 81 - 5 = 76
   \]

4. For \( i = 4 \), the expression is: 
   \[
   3(4^3) - 5 = 3(64) - 5 = 192 - 5 = 187
   \]

5. For \( i = 5 \), the expression is: 
   \[
   3(5^3) - 5 = 3(125) - 5 = 375 - 5 = 370
   \]

6. For \( i = 6 \), the expression is: 
   \[
   3(6^3) - 5 = 3(216) - 5 = 648 - 5 = 643
   \]

7. For \( i = 7 \), the expression is: 
   \[
   3(7^3) - 5 = 3(343) - 5 = 1029 - 5 = 1024
   \]

8. For \( i = 8 \), the expression is: 
   \[
   3(8^3) - 5 = 3(512) - 5 = 1536 - 5 = 1531
   \]

9. For \( i = 9 \), the expression is: 
   \[
   3(9^3) - 5 = 3(729) - 5 = 2187 - 5 = 2182
   \]

10. For \( i = 10 \), the expression is: 
   \[
   3(10^3) - 5 = 3(1000) - 5 = 3000 - 5 = 2995
   \]

Now, let's compute the total sum of all these values:

\[
-2 + 19 + 76 + 187 + 370 + 643 + 1024 + 1531 + 2182 + 2995
\]

I'll calculate the result.The total sum of the expression \( \sum_{i=1}^{10} (3i^3 - 5) \) is:

\[
9025
\]

Would you like more details or have any other questions?

Here are 5 related questions to expand your understanding:

1. What happens if we change the limits of the summation to a different range, say from 1 to 15?
2. How would the result change if the cubic term \( i^3 \) were replaced with \( i^2 \) (a quadratic term)?
3. How do we calculate the sum of cubes from 1 to \( n \), and can that help simplify summations like this?
4. What other mathematical operations can be easily performed using summation notation?
5. How does the general behavior of cubic functions influence the growth of the sum?

**Tip:** When dealing with summations involving powers of \( i \), patterns often emerge for specific ranges of \( i \), which can sometimes lead to simplifications using known formulas.

Evaluate the summation: \( \sum_{i=1}^{10} (3i^3 - 5) \)

Learn how to evaluate the summation \( \sum_{i=1}^{10} (3i^3 - 5) \) with a detailed step-by-step solution. This problem covers key algebraic concepts including summation notation and cubic functions. Suitable for students in Grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation