Which formula is this n n 1 )( 2n 1 )/ 6?

This is the formula for sum of squares – the sum of the squares of the first n counting numbers is n(n+1)(2n+1)/6.

Which of the following is the base case for 4n 1 n 1/2 where n 2?

Which of the following is the base case for 4n+1 > (n+1)2 where n = 2? Explanation: Statement By principle of mathematical induction, for n=2 the base case of the inequation 4n+1 > (n+1)2 should be 64 > 9 and it is true. 10.

Which of the following can be proved by mathematical induction?

Mathematical induction can be used to prove that an identity is valid for all integers n≥1. Here is a typical example of such an identity: 1+2+3+⋯+n=n(n+1)2. More generally, we can use mathematical induction to prove that a propositional function P(n) is true for all integers n≥1.

What is the use of mathematical induction in real life?

Mathematical induction is generally used to prove that statements are true of all natural numbers. The usual approach is first to prove that the statement in question is true for the number 1, and then to prove that if the statement is true for one number, then it must also be true of the next number.

What is the next step in mathematical induction?

The next step in mathematical induction is to go to the next element after k and show that to be true, too: If you can do that, you have used mathematical induction to prove that the property P is true for any element, and therefore every element, in the infinite set.

Why is mathematical induction considered a slippery trick?

Mathematical induction seems like a slippery trick, because for some time during the proof we assume something, build a supposition on that assumption, and then say that the supposition and assumption are both true. So let’s use our problem with real numbers, just to test it out. Remember our property: n 3 + 2 n is divisible by 3.

How do you prove a property by induction?

Proof by Induction. Your next job is to prove, mathematically, that the tested property P is true for any element in the set — we’ll call that random element k — no matter where it appears in the set of elements. This is the induction step. Instead of your neighbors on either side, you will go to someone down the block, randomly,…

When to use the inductive hypothesis in a proof?

Fallacy: In the proof we used the inductive hypothesis to conclude max {a − 1, b − 1} = n 㱺 a − 1 = b − 1. However, we can only use the inductive hypothesis if a − 1 and b − 1 are positive integers.