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
allochka39001 [22]
3 years ago
11

81 POINTS

Mathematics
1 answer:
Jobisdone [24]3 years ago
7 0

Base Case: plug in n = 1 (the smallest positive integer)

If n = 1, then 3n-2 = 3*1-2 = 1. Square this and we see that (3n-2)^2 = 1^2 = 1

On the right hand side, plugging in n = 1 leads to...

n*(6n^2-3n-1)/2 = 1*(6*1^2-3*1-1)/2 = 1

Both sides are 1. So that confirms the base case.

-------------------------------

Inductive Step: Assume that

1^2 + 4^2 + 7^2 + ... + (3k-2)^2 = k*(6k^2-3k-1)/2

is a true statement for some positive integer k. If we can show the statement leads to the (k+1)th case being true as well, then we will have sufficiently proven the overall statement to be true by induction.

1^2 + 4^2 + 7^2 + ... + (3k-2)^2 = k*(6k^2-3k-1)/2

1^2 + 4^2 + 7^2 + ... + (3k-2)^2 + (3(k+1)-2)^2 = (k+1)*(6(k+1)^2-3(k+1)-1)/2

k*(6k^2-3k-1)/2 + (3(k+1)-2)^2 = (k+1)*(6(k^2+2k+1)-3(k+1)-1)/2

k*(6k^2-3k-1)/2 + (3k+3-2)^2 = (k+1)*(6k^2+12k+6-3k-3-1)/2

k*(6k^2-3k-1)/2 + (3k+1)^2 = (k+1)*(6k^2+9k+2)/2

k*(6k^2-3k-1)/2 + 9k^2+6k+1 = (k+1)*(6k^2+9k+2)/2

(6k^3-3k^2-k)/2 + 2(9k^2+6k+1)/2 = (k*(6k^2+9k+2)+1(6k^2+9k+2))/2

(6k^3-3k^2-k + 2(9k^2+6k+1))/2 = (6k^3+9k^2+2k+6k^2+9k+2)/2

(6k^3-3k^2-k + 18k^2+12k+2)/2 = (6k^3+9k^2+2k+6k^2+9k+2)/2

(6k^3+15k^2+11k+2)/2 = (6k^3+15k^2+11k+2)/2

Both sides simplify to the same expression, so that proves the (k+1)th case immediately follows from the kth case

That wraps up the inductive step. The full induction proof is done at this point.

You might be interested in
Please help me it’s math iready for a 7th grader
Artemon [7]

Answer:

m - 7.50 = 12.50

and

m = 7.50 + 12.50

Step-by-step explanation:

The best answers would be:

m - 7.50 = 12.50

and

m = 7.50 + 12.50

First of all, let's figure out what the question is asking us:

The question is asking us to find the amount of money on the gift card

It tells us to use "m" for "the amount of money on the gift card"

Now that we know the following, we can put together the equation

m = the amount of money on the gift card

$7.50 = cost of the book

$12.50 = amount remaining

Since he receives the gift card and then buys a book that costs $7.50, we are going to "have to subtract $7.50 from m". And we know that $12.50 is remaining, so we "make the equation equal to 12.50".

So, let's set up the equation:

m - 7.50 = 12.50

Now that we know the equation, we can solve for "m"

m - 7.50 = 12.50

Step 1: Add 7.50 to both sides.

m − 7.50 7.50 = 12.50 + 7.50

Step 2: Simplify

Answer: m = 20

Another way you can set up this equation is:

m = 7.50 + 12.50

Because we want to find "m", we set it equal to the equation. Since Salim bought a book for $7.50 and had $12.50 remaining, we want to find how much he had on the gift card that made him have $12.50 remaining when he bought the book. So, we have to add 7.50 and 12.50, to find how much is on the gift card.

So, the equation "m = 7.50 + 12.50", makes sense when solving for "m"

Hint: Once you have realized that the equation has Like Terms on the same side you must Combine the Like Terms by either adding them together or subtracting them.

Now that we know the equation, let's solve it

m = 7.50 + 12.50

Step 1: Combine like terms

m = (7.50 + 12.50)

m = 20

Since both equations equal $20 and both make sense, we can conclude that these are the best options.

5 0
2 years ago
Anyone knows this answer ?
jok3333 [9.3K]
Rise over run is 6/4
8 0
3 years ago
Plzzzzzzz help ill give brainliest and there is no answer choice
Iteru [2.4K]

Step-by-step explanation:

Q.4,1

R.3,2

S 1,0

T.-2. 2

(they may be in the wrong way)

Hope this helps

3 0
3 years ago
Read 2 more answers
Which of the following segments is an angle bisector?<br><br> 1. AL<br> 2. JB<br> 3. KC
Orlov [11]

Answer:

I think it's 3) KC

Hopefully it's correct

4 0
3 years ago
How are vectors used in real life?
aleksandr82 [10.1K]

Answer:

Vectors can be used in sports to define how a baseball player moves to catch a high fly baseball they can also be used with aircrafts.


8 0
3 years ago
Read 2 more answers
Other questions:
  • A basket contains one giraffe (G), one panda (P), one monkey (M), and one teddy bear (T). Kelly randomly picks one toy from the
    8·2 answers
  • the volume of a cylinder is given by the formula v=3.14r^2h, where r is the radius of the cylinder and h is the height. suppose
    13·1 answer
  • Choose the polynomial that is written in standard form. (1 point)
    9·1 answer
  • Suppose the probability of rain is 25%
    14·1 answer
  • A bag contains 26 tiles, each with a different letter of the alphabet written on it. You choose 3 tiles from the bag without loo
    9·1 answer
  • What is 16a + 12b in factorise form
    6·1 answer
  • an electrician charges a fee of $50 plus $35 per hour. let y be the cost in dollars of using the electrician for x hours. write
    11·1 answer
  • A cone has a volume of 60. If the radius is 2 feet,
    6·1 answer
  • Α + 4/5 = 3<br> Solve for a to make the proportion true
    11·2 answers
  • The cost of playing pool increases with the amount of time using the table. Identify the independent and dependent quantity in t
    5·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!