# Sn: 1 ∙ 2 + 2 ∙ 3 + 3 ∙ 4 + . . . + n(n + 1) = [n(n + 1)(n + 2)]/3

2 months ago

## Solution 1

Guest #6877546
2 months ago
Steps to inductive
1. show it's true for n=1
2. replace n with k
3. replace k with k+1 and show it is true statement
4. state that it's true

1. show it's true for n=1
1(1+1)=[1(1+1)(1+2)]/3
1(2)=[(2)(3)]/3
2=6/3
2=2
true

2. assume it's true for n=k
1*2+2*3+3*4+...+k(k+1)=[k(k+1)(k+2)]/3

3. show it's true for replaceing k with k+1
1*2+2*3+3*4+...+k(k+1)+(k+1)((k+1)+1)=[(k+1)((k+1)+1)((k+1)+2)]/3
remember from before that 1*2+2*3+3*4+...+k(k+1)=[k(k+1)(k+2)]/3
[k(k+1)(k+2)]/3+(k+1)(k+2)=[(k+1)(k+2)(k+3)]/3
use algebra to show this is true (get it to k=k or something like that)
here's the fun part
deal with left side to get to like right side
[k(k+1)(k+2)]/3+(k+1)(k+2)
make over 3, so muliptlu 2nd thing by 3/3
[k(k+1)(k+2)]/3+((k+1)(k+2)3)/3
[k(k+1)(k+2)+3(k+1)(k+3)]/3
we can undistribute the (k+1)(k+2) part
[(k+1)(k+2)(k+3)]/3
now remember the right side?
[(k+1)(k+2)(k+3)]/3=[(k+1)(k+2)(k+3)]/3
true

4. state it
therefor 1 ∙ 2 + 2 ∙ 3 + 3 ∙ 4 + . . . + n(n + 1) = [n(n + 1)(n + 2)]/3 is true for all real numbers

## 📚 Related Questions

Question
Which expression is equivalent to the given expression? 8(3/4a)−8 1. 3/4a 2. 6a 3. 8(3/4a - 1) 4. 3/4a -1
Solution 1

Step-by-step explanation:

8 (3/4a) - 8

The above expression can be written as:

Since 8 is a common factor here which is multiple of 3/4a as well as 1 therefore taking 8 as a common factor from whole equation,

Therefore option (3) is the correct answer.

Solution 2
The answer is 3 cause we can factor 8 from the whole equasion .
that means to take 8 out of every given data so
since we have 8 multiple 3/4a we can have an 8 and we also have -8 so we can divide it to 8 and take 8 out of it which leads us to 8(3/4a-1)
Question
When a nonzero rational and an irrational number are multiplied, is the product rational or irrational?
Solution 1
The sum of a rational number and an irrational number is irrational.
Question
Harvey is 3 times as old as Jane. The sum of their ages is 48 years. Find the age of each. Jane is a0 years old. Harvey is a1 years old.
Solution 1
Let jane = j

Harvey = 3*j = 3j

Sum = 48 years

j + 3j = 48

4j = 48

j = 48/4

j = 12

Jane, j = 12 years old

Harvey, 3j = 3*12 = 36 years old

Hope this helped.
Solution 2
Harvey is 36, jane is 12

Question
When calculating the effective rate of a loan, which statement or statements must be true if n is greater than 1? I. The length of the loan is greater than a single year. II. The effective rate will exceed the nominal rate. III. The interest will be compounded monthly.
Solution 1

When calculating the effective rate of a loan, the statement or statements must be true if n is greater than 1 is The length of the loan is greater than a single year.
Solution 2

Step-by-step explanation:

its a

Question
All equilateral triangles are also isosceles triangles. TRUE OR FALSE?
Solution 1

false kljfoidjfdiojfoidjfoijfoidjfodijfoidjf

Solution 2

I think it's true but not 100% sure :)

Step-by-step explanation:

Question
Which expression is equivalent to the given expression? −(2n−6) 1. −2(n−3) 2. −2(n−6) 3. 2n−62 4. 2n + 6
Solution 1
−(2n−6)
= -2n + 6
= -2(n - 3)

1. −2(n−3)

Question
Audrey is buying a new car for $32,998.00. She plans to make a down payment of$4,200.00. If she's to make monthly payments of $525 for the next five years, what APR has she paid? Solution 1 Answer: 3.7% Step-by-step explanation: Given: Cost =$32,998

Down payment = $4,200 Remaining amount =$32,998 - $4,200 =$28,798

Monthly payment = $525 Duration = 5 years Present value = Monthly payment × here, k = 12 when compounded monthly r is the rate of interest thus,$28,798 = \$525 ×

or

54.85 =

or

r = 3.7%

Solution 2
For the next five years Audrey paid 3.7% i belive

Question
Simplify. 9y+11z+7y−4z
Solution 1
16y+7z its easy, combine like terms
Solution 2
9y+11z+7y−4z
=16y + 7z

....................
Question
If a = 14 and b = 14, then a = b. What algebraic property is illustrated above? A. identity property of multiplication B. symmetric property C. addition property of equality D. transitive property
Solution 1
B symmetrical property. its pretty easy symmetrical means same so they are the same. think logically :)
Solution 2

its actually D. transitive property

Step-by-step explanation:

Question
Which expression is equivalent to 3x+10−x+123x+10−x+12 ? 1 . 4x + 22 2. 24x 3. 2x + 22 4. 26x
Solution 1
I think its 2 I'm not 100% though

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