Question

Explain mathematical induction step by step in the simplest way possible. Solve the problem Pn: 1+2+2²+2³+...+2ⁿ-1=2n.

a) Start with the base case.
b) Assume the formula is true for some k.
c) Prove it for k+1 using the assumption.
d) Repeat the process for all natural numbers.

Answer 1

Mathematical induction works by proving a base case for the first natural number, assuming a formula is true for an arbitrary case 'k', and then proving it for 'k+1', establishing the formula's validity for all natural numbers.

Step-by-step explanation:

Mathematical induction is a method of proving that a formula or statement is true for every positive integer. It consists of two steps: the base case and the inductive step.

1. Base case: Verify that the statement is true for the first number in the natural numbers, usually n=1.
2. Assume the statement is true for some number k. This is the induction hypothesis.
3. Using the induction hypothesis, prove that the statement is true for k+1.
4. Conclude that since the statement is true for n=1 (base case) and true for k implies it is true for k+1 (inductive step), the statement must be true for all positive integers.

For the problem Pn: 1+2+22+23+...+2n-1 = 2n - a:

1. Base case (n=1): 1 = 21 - 1, which is true.
2. Inductive hypothesis: Assume for some k, 1+2+22+...+2k-1 = 2k - a.
3. Inductive step: Show for k+1, 1+2+22+...+2k-1+2k = 2k+1 - a. Using the inductive hypothesis and algebra, we demonstrate the statement holds for k+1.
4. Conclusion: Thus, by the principle of mathematical induction, the formula is true for all natural numbers n.