Explore induction arguments for five predefined summation identities: integers, odd numbers, squares, cubes and geometric series. Calculate the sum at a chosen positive integer n and read the general base case and inductive step in Calculation details. Numerical checks at one n are illustrations, not proofs for every n.

1 + 2 + … + n

Checks at a chosen n illustrate an identity; they do not prove it for every n. See Calculation details for the general induction argument.

Whole number from 1 to 10,000.

Related Calculators

Induction Proof Formula

For the sum of the first n integers, the following recurrence illustrates the next step. The other identities use their corresponding next term.

P(k + 1) = P(k) + (k + 1)
  • Where P(1) = 1 is the base case for the integer-sum identity
  • k is the current step index
  • P(k) = k(k+1)/2 is the induction hypothesis for an arbitrary positive integer k
  • P(k+1) is the result of the inductive step computed as P(k) + (k+1)

To prove the integer-sum identity, verify P(1) = 1 and show for an arbitrary positive integer k that k(k+1)/2 + (k+1) = (k+1)(k+2)/2. Checking only a selected numerical k does not establish the general step.

What is an Induction Proof?

Definition:

An induction proof is a method of mathematical proof used to demonstrate that a statement holds for all natural numbers by verifying a base case and proving that if it holds for an arbitrary case, it must also hold for the subsequent case.

How to Use the Induction Proof Calculator?

Example Problem:

The following numerical example illustrates the integer-sum identity. Select First n integers, enter n = 5, and select Calculate. The general induction argument appears in Calculation details.

First, verify the base case. In this example, P(1) is set to 1.

Next, the calculator gives P(5) = 15. This is a numerical illustration of the general hypothesis P(k) = k(k+1)/2.

Finally, add the next term to illustrate the step from 5 to 6:

P(6) = P(5) + (5+1)

P(6) = 15 + 6

P(6) = 21

FAQ

What is the purpose of an induction proof?

Induction proofs are used to show that a statement holds for all natural numbers by establishing a base case and proving that if it holds for one case, it holds for the next.

Why is the base case important in an induction proof?

The base case provides the foundation of the proof; without a valid base case, the entire inductive argument is compromised.

How can I ensure my inductive step is correctly formulated?

To ensure the inductive step is correct, clearly state your induction hypothesis and then demonstrate that the case for k+1 follows logically using the established formula.