1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
Ilya [14]
3 years ago
5

Let x,y \epsilon R. Use mathmatical induction to prove the identity.

Mathematics
1 answer:
mariarad [96]3 years ago
8 0

Step-by-step explanation:

We will prove by mathematical induction that, for every natural n,  

(x-y)(x^{n}+x^{n-1}y+...+xy^{n-1}+y^{n})=x^{n+1}-y^{n+1}

We will prove our base case (when n=1) to be true:

Base case:

(x-y)(x^{n}+x^{n-1}y+...+xy^{n-1}+y^{n})=(x-y)(x^{1}+y^{1})=x^2-y^2=x^{1+1}-y^{1+1}

Inductive hypothesis:  

Given a natural n,  

x^{n+1}-y^{n+1}=(x-y)(x^{n}+x^{n-1}y+...+xy^{n-1}+y^{n})

Now, we will assume the inductive hypothesis and then use this assumption, involving n, to prove the statement for n + 1.

Inductive step:

Observe that, for y=0 the conclusion is clear. Then we will assume that y\neq 0.

(x-y)(x^{n+1}+x^{n}y+...+xy^{n}+y^{n+1})=(x-y)y(\frac{x^{n+1}}{y}+x^{n}+...+xy^{n-1}+y^{n})=(x-y)y(\frac{x^{n+1}}{y})+(x-y)y(x^{n}+...+xy^{n-1}+y^{n})=(x-y)y(\frac{x^{n+1}}{y})+y(x^{n+1}-y^{n+1})=(x-y)x^{n+1}+y(x^{n+1}-y^{n+1})=x^{n+2}-yx^{n+1}+yx^{n+1}-y^{n+2}=x^{n+2}-y^{n+2}\\

With this we have proved our statement to be true for n+1.    

In conlusion, for every natural n,

(x-y)(x^{n}+x^{n-1}y+...+xy^{n-1}+y^{n})=x^{n+1}-y^{n+1}

You might be interested in
Please answer the question please. The question is down below.
nekit [7.7K]

Answer:

Please add a picture!

Step-by-step explanation:

7 0
3 years ago
The distance from the school to the library is 12 miles. The distance from the school to the park is 75% of this distance. What
Yakvenalex [24]
9 miles, you dont even need much math, just common sense. divide 12 by 4 and you get three. multiply 3 by 3 to get 9. 
6 0
3 years ago
Read 2 more answers
To find the volume of a box you multiply the links times the width times the height which what is the volume in the cubic feet o
Naya [18.7K]
3*8*16=384 so 384ft^3

5 0
3 years ago
Your brother is 1/5 your age. Your sister is 9 years older than your brother. Your sister is 11 years old. Write and solve an eq
Papessa [141]
Brother=b
b+9=11
b=2
2=1/5a
10=a
a=10

The answer is 10

Hope this helps :)
5 0
3 years ago
The height, in inches, of a randomly chosen American woman is a normal random variable with mean μ = 64 and variance 2 = 7.84. (
Schach [20]

Answer:

Step-by-step explanation:

Given that the height in  inches, of a randomly chosen American woman is a normal random variable with mean μ = 64 and variance 2 = 7.84.

X is N(64, 2.8)

Or Z = \frac{x-64}{2.8}

a)  the probability that the height of a randomly chosen woman is between 59.8 and 68.2 inches.

=P(59.8

b) P(X\geq 59)\\= P(X\geq -1.78)\\ \\=0.9625

c) For 4 women to be height 260 inches is equivalent to

4x will be normal with mean (64*4) and std dev (2.8*4)

4x is N(266, 11.2)

P(4x>260)= \\P(Z\geq -0.53571)\\=0.7054

d) Z is N(0,1)

E(Z19) = P(Z>19)\\= 0.000

since normal distribution is maximum only between 3 std deviations form the mean on either side.

3 0
3 years ago
Other questions:
  • An item is priced at $13.76. If the sales tax is 7%, what does the item cost including sales tax?HURRY
    10·1 answer
  • Help this is due today!!!!
    11·1 answer
  • a $1,000 savings account earns 2.3% interest compounded monthly. how long will it take to double the original investment?
    8·1 answer
  • F(x)=2x+1,g(x)=X2-2x-4
    14·1 answer
  • Maria wants to cover the Box shown below with paper which of the following could be the dimensions of the paper so that Maria ha
    13·1 answer
  • The sum of five more than a certain number and 10 more than twice the number is equal to the product of 2 and the number increas
    7·1 answer
  • Eliminate the parameter and write the corresponding rectangular equation whose graph represents the curve.
    8·1 answer
  • A city planner is rerouting traffic in order to work on a stretch of road. The equation of the path of the old route can be desc
    15·1 answer
  • Divide. Enter the quotient in simplest form. 7 10 ÷ 2 5 =
    9·2 answers
  • PLEASE HELP I WILL MARK YOU BRAINLIEST IF CORRECT
    13·2 answers
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!