need help with mathematical induction?
For a and b positive real numbers, let P (n) denote that (a+b)^n ≥ a^n +b^n. Prove by mathematical induction that for P (n) is true for n any positive integer.
1. Clearly state the predicate P(n). This is a statement not a number
2. Basis step. show that P (n0) is true, usually n0 = 0 or n0 = 1. This step is usually simple, yet veryimportant. Without this step, there is no basis for the induction part
3. inductive step: here you can use either of the two mathematical induction principles
- BrambleLv 79 years agoBest Answer
It's not too clear what help you want. You seem to know already what induction is – you describe it quite well. Do you just want to see it applied to the given problem? This is what I'll do:
1. Prove (a+b)ⁿ ≥ aⁿ +bⁿ for all n from 1 to ∞ (note zero is not a positive integer)
2. For n=1, LHS = (a+b) and RHS = a+b so the formula holds and P(n=0) is true.
3. For n = n > 1, LHS = (a+b)ⁿ = aⁿ + bⁿ + other positive terms. RHS = aⁿ + bⁿ.
So LHS ≥ RHS for all n≥1, so P(n) is verified.