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14. Prove that for every positive integer n, (n- 1)2"+1 + 2. %3D

14. Prove that for every positive integer n, (n- 1)2"+1 + 2. %3D

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Sure! Here's the transcription of the text and a description of the image suitable for an educational website: --- ### Mathematical Proofs **14. Statement to Prove:** Prove that for every positive integer \( n \), the sum \(\sum_{k=1}^n k2^k = (n-1)2^{n+1} + 2\). **32. Statement to Prove:** Prove that 3 divides \( n^3 + 2n \) whenever \( n \) is a positive integer. **34. Statement to Prove:** Prove that 6 divides \( n^3 - n \) whenever \( n \) is a nonnegative integer. --- Each statement requires proving a mathematical relationship using well-known mathematical principles and theorems. No graphs or diagrams are present in the image. The focus is on algebraic manipulation and number theory.
Transcribed Image Text:Sure! Here's the transcription of the text and a description of the image suitable for an educational website: --- ### Mathematical Proofs **14. Statement to Prove:** Prove that for every positive integer \( n \), the sum \(\sum_{k=1}^n k2^k = (n-1)2^{n+1} + 2\). **32. Statement to Prove:** Prove that 3 divides \( n^3 + 2n \) whenever \( n \) is a positive integer. **34. Statement to Prove:** Prove that 6 divides \( n^3 - n \) whenever \( n \) is a nonnegative integer. --- Each statement requires proving a mathematical relationship using well-known mathematical principles and theorems. No graphs or diagrams are present in the image. The focus is on algebraic manipulation and number theory.
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