Mathematical Induction

What is the third step in Mathematical induction?

• P(1)

• P(k+1)

• P(k)

• n=k

1 + 3 + 5 + 7 + . . . + (2n − 1) = n²

What must we show first?

• The statement is true for n = 1:

  (2)(1) − 1 = 1²

• The statement is true for n = k:

  1 + 3 + 5 + 7 + . . . + (2k − 1) = k²

• The statement is true for n = k + 1:

  1 + 3 + 5 + 7 + . . . + (2k − 1) + (2k + 1) = (k + 1)²

Tags

CCSS.HSA.APR.C.4

CCSS.HSA.REI.A.1

CCSS.HSA.SSE.A.1

CCSS.HSA.SSE.A.2

CCSS.HSA.SSE.B.3

Let P(n) = 2n − 1. Evaluate:

a) P(k)

b) P(k + 1)

• a) P(k) = 2k − 1

  b) P(k + 1) = 2n + 1

• a) P(k) = 2k + 1

  b) P(k + 1) = 2(k + 1) - 1

• a) P(k) = 2k − 1

  b) P(k + 1) = 2k + 1

Tags

CCSS.HSF.IF.A.2

1 + 3 + 5 + 7 + . . . + (2n − 1) = n²

To prove this by mathematical induction, what will be the induction assumption?

• The statement is true for n = k:

  1 + 3 + 5 + 7 + . . . + (2k − 1) = k²

• The statement is true for n = 1:

  (2)(1) − 1 = 1²

• The statement is true for n = k + 1:

  1 + 3 + 5 + 7 + . . . + (2k − 1) + (2k + 1) = (k + 1)²

What is the first step in Mathematical Induction?

• P(k)

• n=k

• P(k+1)

• P(1)

Which of these is the first step in mathematical induction?

• Prove the statement is true for the first element in the set.

• Show that if the statement is true for the first k elements, then it is true for the (k+1)st case.

• Prove that the problem you are working on is the base to all proofs.

• None of these are correct.

Which of the following is the induction step in mathematical induction?

• Show that if the statement is true for the first k elements, then it is true for the (k+1)st element in the set.

• Show that the statement is true for the first few elements in the set.

• Show that your math problem is different from all other math problems.

• None of these are correct.

1 + 3 + 5 + 7 + . . . + (2n − 1) = n²
On the basis of this assumption,
[The statement is true for n = k:
1 + 3 + 5 + 7 + . . . + (2k − 1) = k²]
What must we show?

• The statement is true for n = 1:
  2x1 − 1 = 1²
• The statement is true for n = k:
  1 + 3 + 5 + 7 + . . . + (2k − 1) = k²
• The statement is true for n = k + 1:
  1 + 3 + 5 + 7 + . . . + (2k − 1) + (2k + 1) = (k + 1)²

Tags

CCSS.HSA.APR.C.4