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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}

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padilas [110]

<u>Unit rate 4:</u>

The table

d      n

4       1

8       2

16      4

<u>Unit rate 1/4:</u>

Table

d      n

1       4

4       16

16      64

and

The equation n=4d

<u>Unit rate 16:</u>

Equation:

d = 16n

Step-by-step explanation:

The unit in the unit rate is dollars per unit ounces which means that the unit rate calculated by

Unit\ rate = \frac{dollars}{ounces}

So,

The first option is the equation:

d = 16n\\\frac{d}{n} = 16

The unit rate is 16 dollars per ounce

The second option is the table:

d      n

1       4

4       16

16      64

We can take any pair of values of d and n from the table to calculate the unit rate

So taking

d = 1

n=4

Unit\ rate = \frac{d}{n}\\= \frac{1}{4}

The unit rate for table is: 1/4 dollars per ounces

Third option is the equation:

n = 4d

n = 4d

dividing both sides by n

\frac{n}{n} = \frac{4d}{n}\\1 = \frac{4d}{n}

dividing both sides by 4

\frac{1}{4} = \frac{4d}{4n}\\\frac{d}{n} = \frac{1}{4}

the unit rate is 1/4 dollars per ounce

Fourth option is the table:

d      n

4       1

8       2

16      4

We will take any pair of d and n to find the unit rate

So,

Taking

d = 4

n =1

Unit\ rate = \frac{d}{n} \\= \frac{4}{1}\\=4

The unit rate is 4 dollars per unit ounce.

Keywords: Unit rate, units

Learn more about unit rate at:

  • brainly.com/question/4550858
  • brainly.com/question/4639731

#LearnwithBrainly

7 0
2 years ago
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