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1. Use strong mathematical induction to show that any postage of at least 18¢ can be obtained using 4¢ and 7¢ stamps.

1. Use strong mathematical induction to show that any postage of at least 18¢ can be obtained using 4¢ and 7¢ stamps.

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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**Problem Statement:** 1. Use **strong mathematical induction** to show that any postage of at least 18¢ can be obtained using 4¢ and 7¢ stamps. --- **Explanation:** This problem requires the use of strong mathematical induction, a powerful method often used in mathematical proofs. The goal is to demonstrate that for any amount of postage that is 18 cents or more, it is possible to achieve this exact amount using combinations of stamps valued at 4 cents and 7 cents. **Steps for Induction:** 1. **Base Cases:** Check small examples such as 18¢, 19¢, 20¢, and 21¢ to see if they can be formed using just 4¢ and 7¢ stamps. 2. **Inductive Step:** Assume for some integer \( k \) that all postage values from 18¢ up to \( k \) cents can be formed using 4¢ and 7¢ stamps. Then show that \( k+1 \) cents can also be formed. 3. **Conclusion:** By proving the base cases and the inductive step, you conclude that the statement holds for all postage values of 18¢ and above. Imagine preparing this proof for educational purposes, highlighting how the combination of different values within a set can cover a vast range of possibilities through structured reasoning.
Transcribed Image Text:**Problem Statement:** 1. Use **strong mathematical induction** to show that any postage of at least 18¢ can be obtained using 4¢ and 7¢ stamps. --- **Explanation:** This problem requires the use of strong mathematical induction, a powerful method often used in mathematical proofs. The goal is to demonstrate that for any amount of postage that is 18 cents or more, it is possible to achieve this exact amount using combinations of stamps valued at 4 cents and 7 cents. **Steps for Induction:** 1. **Base Cases:** Check small examples such as 18¢, 19¢, 20¢, and 21¢ to see if they can be formed using just 4¢ and 7¢ stamps. 2. **Inductive Step:** Assume for some integer \( k \) that all postage values from 18¢ up to \( k \) cents can be formed using 4¢ and 7¢ stamps. Then show that \( k+1 \) cents can also be formed. 3. **Conclusion:** By proving the base cases and the inductive step, you conclude that the statement holds for all postage values of 18¢ and above. Imagine preparing this proof for educational purposes, highlighting how the combination of different values within a set can cover a vast range of possibilities through structured reasoning.
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