assessments/760206/variants/760206/take/12/ SKETCHPAD Question 13 Use mathematical induction to prove the following: Σ" ³= 1³ +2³+3³ + ... +n³ = a Note: If you would rather show your work on a separate sheet of paper load your work in Question 14. PENCIL n²(n + 1)² 4 THIN ✔ 2 ☆ BLACK V
assessments/760206/variants/760206/take/12/ SKETCHPAD Question 13 Use mathematical induction to prove the following: Σ" ³= 1³ +2³+3³ + ... +n³ = a Note: If you would rather show your work on a separate sheet of paper load your work in Question 14. PENCIL n²(n + 1)² 4 THIN ✔ 2 ☆ BLACK V
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.5: The Binomial Theorem
Problem 37E
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![### Question 13: Mathematical Induction Proof
**Problem Statement:**
Use mathematical induction to prove the following equation:
\[
\sum_{i=1}^{n} i^3 = 1^3 + 2^3 + 3^3 + \ldots + n^3 = \frac{n^2(n + 1)^2}{4}
\]
**Instructions:**
Note: If you would rather show your work on a separate sheet of paper, please upload your work in Question 14.
#### Explanation:
To prove this equation using mathematical induction, you will need to follow these steps:
1. **Base Case**: Show that the statement holds true for \( n = 1 \).
2. **Induction Hypothesis**: Assume that the statement is true for some arbitrary positive integer \( k \). That is, assume:
\[
\sum_{i=1}^{k} i^3 = \frac{k^2(k + 1)^2}{4}
\]
3. **Induction Step**: Prove that if the statement is true for \( n = k \), then it is also true for \( n = k + 1 \).
\[
\sum_{i=1}^{k+1} i^3 = \sum_{i=1}^{k} i^3 + (k+1)^3
\]
4. **Conclusion**: By the principle of mathematical induction, conclude that the statement holds for all positive integers \( n \).
### Additional Information:
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The sketchpad tool includes options for a pencil with different thickness settings (currently set to "Thin") and color options (currently set to "Black").
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Transcribed Image Text:### Question 13: Mathematical Induction Proof
**Problem Statement:**
Use mathematical induction to prove the following equation:
\[
\sum_{i=1}^{n} i^3 = 1^3 + 2^3 + 3^3 + \ldots + n^3 = \frac{n^2(n + 1)^2}{4}
\]
**Instructions:**
Note: If you would rather show your work on a separate sheet of paper, please upload your work in Question 14.
#### Explanation:
To prove this equation using mathematical induction, you will need to follow these steps:
1. **Base Case**: Show that the statement holds true for \( n = 1 \).
2. **Induction Hypothesis**: Assume that the statement is true for some arbitrary positive integer \( k \). That is, assume:
\[
\sum_{i=1}^{k} i^3 = \frac{k^2(k + 1)^2}{4}
\]
3. **Induction Step**: Prove that if the statement is true for \( n = k \), then it is also true for \( n = k + 1 \).
\[
\sum_{i=1}^{k+1} i^3 = \sum_{i=1}^{k} i^3 + (k+1)^3
\]
4. **Conclusion**: By the principle of mathematical induction, conclude that the statement holds for all positive integers \( n \).
### Additional Information:
At the bottom of the screen, there are progress indicators showing that 10 out of 14 total questions have been answered. The current weather is displayed as 76°F with haze.
The sketchpad tool includes options for a pencil with different thickness settings (currently set to "Thin") and color options (currently set to "Black").
Make sure to save any changes and upload your work if you choose to use a separate sheet for Question 14.
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