- A statement can be proved using PMI:
Proving Series:
- for a statement involving a series, e.g.
1+3+5+\dots+(2n-1)=n^{2}
where the last statement is the nth index of the series it can be proved using mathematical induction ### Process: Prove using mathematical induction that: $1+3+5+\dots+(2n-1)=n^{2}$ is true for all positive integers n. 1. Prove the statement true for $n=1$ - ![[Pasted image 20231013135212.png]] - proving true for n=1, (the first positive integer for $n\geq 1$ ) 2. assume a statement to be true for $n=k$ (any positive integer) - ![[Pasted image 20231013135950.png]] - the `*` is necessary to reference later 3. prove the statement true for $n=k+1$ (the next index after $n=k$) - ![[Pasted image 20231013140324.png|600]] ### Rational: -