Question

Use mathematical induction to prove the statement is true for all positive integers n, or show why it is false.

4 ⋅ 6 + 5 ⋅ 7 + 6 ⋅ 8 + ... + 4n( 4n + 2) = [4(4n+1)(8n+7) / 6

Answer 1

@uri @skullpatrol @mathmate

Answer 2

Have you done mathematical induction?

Answer 3

I have seen it in my lesson but its extremely vague. Is it simple enough for you to give me a short explanation. Or should I try and figure it out first?

Answer 4

The three steps are:
1. Verify that the statement is true for n=1 by substitution.
2. \(Assuming\) that the statement is true for case n, show that it is also true for case n+1. This is done by adding the (n+1) term on the left hand side and try to manipulate the expression algebraically so that it becomes the expression on the right, with n replaced by (n+1).
3. Conclude that the statement is true for all n >=1.

Answer 5

So for step 1 I just plug in 1 for n and solve?

Answer 6

Or I just solve for n

Answer 7

You don't really "solve".
You put in n=1 to evaluate both sides. Verify that the left hand side equals the right hand side.
If they are not equal, check your question, or check your calculations.

Answer 8

Oh okay! Just a sec while I solve

Answer 9

Example:
show that \(1+2+3+...+n=\frac{n(n+1)}{2}\)
Step 1:
put n=1, LHS=1, RHS=(1*2)/2=1,
So the statement is valid for n=1.

Answer 10

Step 2:
Assume that the statement is true for n,
then
\(1+2+3+...+n=\frac{n(n+1)}{2}\)
for case n+1, we have
\(1+2+3+...+n + (n+1)=\frac{n(n+1)}{2}+(n+1)=\frac{n(n+1)+2n+2}{2}=\frac{(n+1)(n+2)}{2}\)
Therefore the statement is also true for n+1, hence
by mathematical induction, \(1+2+3+...+n=\frac{n(n+1)}{2}\) is true for all n \(\ge\)1.

Answer 11

Crap... I think you lost me already

Answer 12

Have you done the first step?

Answer 13

No.. what are LHS and RHS ?

Answer 14

It's a mathematical jargon to mean left-hand side and right hand side.

Answer 15

Okay. And how did you get that equation you had for RHS?

Answer 16

I think you'd be better off studying your teacher's notes, or study my example to understand how the logic goes before attempting the more difficult problem your teacher gave you.

Answer 17

For step 1? You'd put n=1 into the LHS and RHS and show that they are equal.
If there's no arithmetical error and they are not equal, you just proved that the statement is not true.

Answer 18

Okay. So unless I messed up I got the left term as 24 and the right as 50

Answer 19

Let me check.

Answer 20

That's what I got too, so that means that the statement (that you've written) is not true.
Check that there's no typo.

Answer 21

Looks good to me

Answer 22

If you look just on the left hand side,
4.6+5.7+....
will not make 4n(4n+2) because
then first term is 4.6, second term (n=2) is 8.10, not 5.7
There is definitely a mix-up somewhere.

Answer 23

Right. So at this point I am done with this problem?

Answer 24

The problem should read, if the LHS is good:
\(4.6+5.7+6.8+...+(n+3)(n+5)=\large \frac{n(2n^2+27n+115)}{6}\)

Answer 25

I would contact the teacher if there is no typo.

Answer 26

No the point are multiplication signs

Answer 27

In any case, study the example I gave you. That would help you when you get the correct problem.

Answer 28

Yes, I used the points as multiplication signs, just like the original problem.

Answer 29

And I believe Ive figured out how to complete these problems

Answer 30

Good! That's the whole point of exercise.

Answer 31

Good luck with your studies and assignments.

Answer 32

Awesome! thanks so much!!

Answer 33

You're welcome!