💯 Inequality Proof | Prove 3^n is greater than or equal to 2n + 5 by Mathematical Induction

💯 Receive Comprehensive Mathematics Practice Papers Weekly for FREE

Timeline:
0:00 Intro
0:32 Step 1 Checking the initial value
1:16 Step 2 Assumption
1:30 Step 3 Proof
4:31 Conclusion

Inequality Proof by Mathematical Induction

Mathematical induction proves that a statement is true for all positive integers n greater than or equal to a certain number. To prove that 3^n is greater than or equal to 2n + 5 for all positive integers n greater than or equal to 2, you would perform the following steps:

Show the base case: Prove that the statement is true for the smallest value of n that satisfies the conditions, typically n = 2. In this case, 3^2 = 9, which is greater than or equal to 2 * 2 + 5 = 9.

Assume the statement is true for n = k: Assume that 3^k is greater than or equal to 2k + 5 for some positive integer k.

Prove that the statement is true for n = k + 1: Using the assumption, show that 3^(k+1) is greater than or equal to 2(k+1) + 5. To do this, you would use the property that 3^(k+1) = 3 * 3^k. Thus, 3^(k+1) is greater than or equal to 3 * (2k + 5) = 6k + 15. Now, you need to show that 6k + 15 is greater than or equal to 2(k + 1) + 5. Using algebraic manipulation, you can simplify this to 4k is greater than or equal to 2, which is true for all positive integers k.

Conclude: Based on steps 1, 2, and 3, you can conclude that 3^n is greater than or equal to 2n + 5 for all positive integers n greater than or equal to 2.

Following these steps, you have used mathematical induction to prove that 3^n is greater than or equal to 2n + 5 for all positive integers n greater than or equal to 2.