Can a conditional statement be proved by induction?
Proof by Induction The logic associated with conditional statements can be used to prove a property holds for an infinitely large set. For example, 2 n ≥ n + 1 for all natural numbers, let P(n) be the statement, the property hold for the natural number n.
What does it mean to prove a formula?
A mathematical proof shows a statement to be true using definitions, theorems, and postulates. Just as with a court case, no assumptions can be made in a mathematical proof. Every step in the logical sequence must be proven. The premises in the proof are called statements. Proofs can be direct or indirect.
How to prove that a(n) holds for all positive integers n?
Let A(n) be an assertion concerning the integer n. If we want to show that A(n) holds for all positive integer n, we can proceed as follows: Induction basis: Show that the assertion A(1) holds. Induction step: For all positive integers n, show that A(n) implies A(n+1). 3 Standard Example
How do you prove that b(n+1) holds?
Expanding the right hand side yields n3/3 + 3n2/2 + 13n/6 + 1 One easily verifies that this is equal to (n+1)(n+2)(2(n+1)+1)/6 Thus, B(n+1) holds. Therefore, the proof follows by induction on n. 8 Tip How can you verify whether your algebra is correct?
Is -is an identity for n = k+1?
Therefore the result is also true for n = k+ 1. Hence by the principle of mathematical induction, the result is true for all i.e., -is an identity. Noticing that the sum on the left telescopes and distributing the sum over the two terms on the right, we finally have: i.e., -is an identity. 8 clever moves when you have $1,000 in the bank.
Is the sum of -th powers of first natural numbers a polynomial?
If is a natural number than the sum of -th powers of the first natural numbers is a polynominal in variable of degre and with rational coefficients (Wikipedia calls them Faulhaber polynomials, see Faulhaber’s formula – Wikipedia ):