No, you can't just assert this -- you have to show it.It_Angel said:Homework Statement
Prove that 3^n >= 2n+1 for all natural numbers.
Homework Equations
3^n >= 2n+1 [is bigger or equal to]
The Attempt at a Solution
3*1>=2+1
True for n=1
Assumption: 3^k>=2k+1
3^(k+1)>=2k+3
It_Angel said:3^k*3>=2k+3
(2k+1)*3>=2k+3 <---can I just substitute 2k+1 into 3^k as per my assumption, because 3^k is bigger, and (2k+1)*3 is bigger than 2k+3? If so, is the proof complete?
Thanks in advance.
Mark44 said:Use the fact that 3^(k + 1) = 3 * 3k, and use your assumption that 3^i >= 2k + 1.
Proof by induction is a mathematical method used to prove that a statement is true for all natural numbers. It involves three steps: the base case, the inductive hypothesis, and the inductive step.
The base case is used to prove that the statement is true for the first natural number. The inductive hypothesis assumes that the statement is true for a certain natural number, and the inductive step uses this assumption to prove that the statement is also true for the next natural number. By repeating this process, the statement is proven to be true for all natural numbers.
Proof by induction is typically used to prove statements about natural numbers, such as equations, inequalities, and divisibility. It can also be used to prove statements about sets, functions, and graphs.
Proof by induction can only be used to prove statements about natural numbers. It cannot be used for real numbers, irrational numbers, or complex numbers. Additionally, the inductive step must be carefully constructed and may not always be straightforward.
No, proof by induction is not a universal method and may not be applicable to all mathematical statements. Some statements may require other methods of proof, such as direct proof or proof by contradiction.