Use the Principle of Mathematical Induction to show that the statement is true for all natural numbers n.n = 32n

What will be an ideal response?


First we show that the statement is true when n = 1.
 For n = 1, we get 32 = 3(2?1) = 32
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 multiply the next term,  to both sides of the equation.
 k(k + 1) = 32k(k + 1)
 (k + (k + 1)) = 32k
 (2k + 1) = 3(2k + 2k + 2))
 3(2(2k + 1)) = 3(2(2k + 1))
Condition II is satisfied. As a result, the statement is true for all natural numbers n.

Mathematics

You might also like to view...

Find the derivatives of all orders of the function.y = x3 + x2 - 2x + 14

What will be an ideal response?

Mathematics

Provide an appropriate response.A roadway inclines 12.5°. How many feet of roadway must be paved if the hill is 20 ft high? Round to the nearest foot.

Fill in the blank(s) with the appropriate word(s).

Mathematics

Write in logarithmic form.2-2 = 

A. log2 -2 = 
B. log2  = -2
C. log-2  = 2
D. log1/4 2 = -2

Mathematics

Provide an appropriate response.The standard normal probability density function is defined by f(x) = e- x2/2.  (a) Use the fact that dx = 1 to show that dx =.(b) Use the result in part (a) to show that the standard normal probability density function has variance 1.

What will be an ideal response?

Mathematics