Use the Principle of Mathematical Induction to show that the statement is true for all natural numbers n.1 ? 2 + 2 ? 3 + 3 ? 4 + . . . + n(n + 1) = 

What will be an ideal response?


First we show that the statement is true when n = 1.
 For n = 1, we get 2 =  = 2.
This is a true statement and Condition I is satisfied.

Next, we assume the statement holds for some k. That is,
  is true for some positive integer k.
We need to show that the statement holds for k + 1. That is, we need to show that
 
So we assume that  is true and add the next term,  to both sides of the equation.
1 ? 2 + 2 ? 3 + 3 ? 4 + . . . + k(k + 1) + (k + 1)(k + 2) =  + (k + 1)(k + 2)
  + 
 
 
Condition II is satisfied. As a result, the statement is true for all natural numbers n.

Mathematics

You might also like to view...

Answer the given question by setting up and solving the appropriate proportion. If necessary, round the answer to two decimal places.Given that 1.000 kg = 2.205 lb, what mass in kilograms is equivalent to 98.5 lb?

A. 0.02 kg B. 44.67 kg C. 4.47 kg D. 2.03 kg

Mathematics

Estimate. Then find the actual sum0.212 + 70 + 280.1

A. Estimate: 350.4; Actual: 350.312 B. Estimate: 350.3; Actual: 350.312 C. Estimate: 350.3; Actual: 349.312 D. Estimate: 350.4; Actual: 349.312

Mathematics

Solve the problem.Sven can type 48 words per minute. How many words would he type in  hour (30 minutes)?

A. 24 words B. 720 words C. 96 words D. 1,440 words

Mathematics

Solve. Use ? ? 3.14 when necessary.Find the volume of a tin can with radius 6.5 in. and height 5 in. Round your answer to the nearest tenth.

A. 204.1 in.3 B. 102.1 in.3 C. 663.3 in.3 D. 2653.3 in.3

Mathematics