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
FromTheMoon [43]
3 years ago
14

Prove that the statement (ab)^n=a^n * b^n is true using mathematical induction.

Mathematics
2 answers:
IRISSAK [1]3 years ago
6 0

Hello, n being an integer, we need to prove that one statement depending on n is true, let's note it S(n).

The mathematical induction involves two steps:

Step 1 - We need to prove S(1), meaning that the statement is true for n = 1

Step 2 - for k integer > 1, we assume S(k) and we need to prove that S(k+1) is true.

Imagine that you are a painter and you need to paint all the trees on one side of a road. You have several colours that you can use but you are asked to follow two rules:

Rule 1 - You need to paint the first tree in white.

Rule 2 - If one tree is white you have to paint the next one in white too.

What colour do you think all the trees will be painted?

Do you see why this is very important to prove the two steps as well ?

Let's do it in this example.

Step 1 - for n = 1, let's prove that S(1) is true, meaning  (ab)^1=a\cdot b =a^1\cdot b^1

So the statement is true for n = 1

Step 2 - Let's assume that this is true for k, and we have to prove that this is true for k+1

So we assume S(k), meaning that (ab)^k=a^k\cdot b^k

and what about S(k+1), meaning (ab)^{k+1}=a^{k+1}\cdot b^{k+1} ?

We will use the fact that this is true for k,

(ab)^{k+1}=(ab)\cdot (ab)^k =(ab) \cdot a^k \cdot b^k

We can write it because the statement at k is true and then we can conclude.

(ab)^{k+1}=(ab)\cdot (ab)^k =(ab) \cdot a^k \cdot b^k=a^{k+1}\cdot b^{k+1}

In conclusion, we have just proved that S(n) is true for any n integer greater or equal to 1, meaning (ab)^{n}=a^{n}\cdot b^{n}

Hope this helps.

Do not hesitate if you need further explanation.

Thank you

mamaluj [8]3 years ago
5 0

Answer:

see below

Step-by-step explanation:

      (ab)^n=a^n * b^n

We need to show that it is true for n=1

assuming that it is true for n = k;

(ab)^n=a^n * b^n

( ab) ^1 = a^1 * b^1

ab = a * b

ab = ab

Then we need to show that it is true for n = ( k+1)

or (ab)^(k+1)=a^( k+1) * b^( k+1)

Starting with

  (ab)^k=a^k * b^k    given

Multiply each side by ab

ab *  (ab)^k= ab *a^k * b^k

   ( ab) ^ ( k+1) = a^ ( k+1) b^ (k+1)

Therefore, the rule is true for every natural number n

You might be interested in
PLEASE HELP ASAP! WILL GIVE BRAINLIEST TO CORRECT AND STRONG ANSWER! THANK YOU IM STRUGGLINGGG
KengaRu [80]

Answer:

Step-by-step explanation:

a3 = 2

d = -5

=============

a4 = a3 - 5

a4 = 2 - 5

a4 = - 3

===============

a5 = a4 - 5

a5 = -3 - 5

a5 = - 8

Just take a4 as your answer.

7 0
3 years ago
HELP PLEASEEE with steps
zysi [14]
<h3>Answer: Choice C) 421.9</h3>

=======================================

Explanation:

You're on the right track. You wrote down the proper expression to get the final answer, assuming you meant to write 75/4 as the third term inside the parenthesis. This works because each time you cut the side length in half to get each smaller triangle's side. The 3 is because there are 3 sides for each of the triangles. Much of this I have a feeling you already know as you wrote down the expression on the page, though I'm not 100% sure of your mindset. Computing this expression leads to 421.875 which rounds to 421.9

note: an alternative is to write 3*75 + 3*75/2 + 3*75/4 + 3*75/8, though that is more work. It's better to have that 3 factored out.

8 0
3 years ago
\frac{(2_1-4_{2}) ^{4} } {2}<br><br>I will give you brainliest
fenix001 [56]

Answer:

I think the answer is 8

Step-by-step explanation:

8 0
3 years ago
Use long division to find the quotient: 7/8.<br><br> I just don´t get it?
Aleonysh [2.5K]

Answer:

0.875

Step-by-step explanation:

you just divide 7 by 8, and you would get 0.875

7/8=0.875

7 0
3 years ago
A flat surface made up of points that extends indefinitely in all directions
lapo4ka [179]

Good morning Sir/Ma'am!

Answer:

A plane is a flat surface made up of points that extends infinitely in all directions.

Hope this helps! Have a great day!

~Sky

5 0
3 years ago
Other questions:
  • a street that is 116 meters long is covered in snow city workers are using a snow plow to clear the street a tire on the snow pl
    8·1 answer
  • An inequality is shown below. x + 2.5 &lt; 20 What is the greatest value of x from the set {10.5, 12.5, 17.5, 19.5} that makes t
    14·1 answer
  • An item is regularly priced at $25. It is on sale for 80% off the regular price. What is the sale price?
    12·2 answers
  • You measured 6 cups of water and 5 cups of lemon juice to make how many total quarts of lemonade?(note: please covert to quarts
    6·1 answer
  • Which sequence below represents an exponential sequence A.) {2,6,10,14,18,...} B.) {3,5,9,16,24,...} C.) {4,8,24,96,...} D.) {25
    7·1 answer
  • Expand Your answer should be a polynomial in standard form. (x+3)(x−5)
    15·1 answer
  • What would the profit be if the flight had 8 first-class passengers and 42 tourist-class
    14·1 answer
  • The two triangles are similar, What is the length of DE?
    7·1 answer
  • A circle has a radius of 5 centimeters. What is the area?
    14·1 answer
  • Please help thanks.<br><br><br><br><br><br><br><br>.___________.​
    15·2 answers
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!