Question

Use Mathematical Induction to prove the statement is true for all positive integers n, of show why it is false.
1^2+4^2+7^2+...+(3n-2)^2=n(6n^2-3n-1)/2

Answer 1

did you check to see if it was true first ?

Answer 2

I dont know how to do that...

Answer 3

ah ok

Answer 4

Yeah so you see my dilemma

Answer 5

\[\sum_{k=1}^n(3k-2)^2\] is what you have right?

Answer 6

What?? how did you get that?

Answer 7

i wrote your sum in sigma notation is all

Answer 8

oh okay haha so how do i do this problem?

Answer 9

to check that it is indeed true, note that
\[(3k-2)^2=3k^2-12k+4\] so
\[\sum_{k=1}^n(3k-2)^2=3\sum_{k=1}^n k^2 -12\sum_{k=1}^nk +4\sum_{k=1}^n 1\]

Answer 10

you have formulas for each of these three, so it it is algebra from here on in
that is how your prof made up the problem

Answer 11

okay so using algebra how would i prove this? Do i just plug in the numbers to k?

Answer 12

you use the following
\[\sum_{k=1}^nk^2=\frac{n(n+1)(2n+1)}{6}\]
\[\sum_{k=1}^nk=\frac{n(n+1)}{2}\]
\[\sum_{k=1}^n1=n\]

Answer 13

\[\sum_{k=1}^n(3k-2)^2=3\sum_{k=1}^n k^2 -12\sum_{k=1}^nk +4\sum_{k=1}^n 1\]
\[=3\left(\frac{n(n+1)(2n+1)}{6}\right)-12\left(\frac{n(n+1)}{2}\right)+4n\]

Answer 14

or
\[\frac{n(n+1)(2n+1)}{2}-6n(n+1)+4n\] i will let you finish the algebra for your self
then we can prove it is true by induction

Answer 15

I am so confused.

Answer 16

yes i sensed that that was true. lets back up a bit
are you familiar with this sigma notation?

Answer 17

no.

Answer 18

oh okay then lets forget it

Answer 19

\[1+2+3+4+...+n=\frac{n(n+1)}{2}\] how about that formula? have you seen it?

Answer 20

okay...

Answer 21

not before now no

Answer 22

ok then lets forget that also, although that is a very good formula to know
it tells you how to add up the first consecutive \(n\) whole numbers, like
\[1+2+3+...+99+100=\frac{100\times 101}{2}=5050\]

Answer 23

okay i get that formula

Answer 24

however, if you do not know these formulas, then we cannot do step one, which is to show that that
\[1^2+4^2+7^2+...+(3n-2)^2=\frac{n(6n^2-3n-1)}{2}
\] is correct

Answer 25

so lets just assume that it is correct, and proceed to prove it by induction
ready?

Answer 26

Is there any way you could just give me the answer and work backwards to show how you found it?

Answer 27

Okay

Answer 28

first of all you have to check that it is true if \(n=1\)

Answer 29

This equation does not equal eachother

Answer 30

if n=1

Answer 31

so you have to replace \(n\) by 1 on the right side of the equal sign
that is, you have to show that
\[1=\frac{1\times (6\times 1^2-3\times 1-1)}{2}\] is it clear why i wrote that?

Answer 32

yes i get that part

Answer 33

ok and if it does not work, you are done, it is not true
but it does work, do the arithmetic carefully and you will see it is right

Answer 34

well i got 1 on the left side and 2 on the right side

Answer 35

did you remember to divide by 2?

Answer 36

No! so they are equal! what do i do now?

Answer 37

\[1=\frac{1\times (6\times 1^2-3\times 1-1)}{2}=\frac{6-3-1}{2}=\frac{2}{2}=1\]

Answer 38

Okay i get that part. Its pretty easy

Answer 39

ok now comes the tricky part.
we get to "assume it is true if \(n=k\)" so lets replace \(n\) by \(k\) and write what we know, namely that
\[1^2+4^2+7^2+...+(3k-2)^2=\frac{k(6k^2-3k-1)}{2}\]

Answer 40

i hope you see i did almost nothing, just replaced \(n\) by \(k\) everywhere i saw it.
now we have to prove that it is true if \(n=k+1\)
the first thing we need is to know what to write if \(n=k+1\)

have you done a proof by induction before?

Answer 41

No

Answer 42

then this is a really bad one to start with because the algebra is going to suck
have you SEEN a proof by induction before?

Answer 43

no haha and just my luck i guess

Answer 44

hmm, then i think this is not going to make the slightest bit of sense to you
do you have to hand this in? is it for a class?

Answer 45

yeah its a worksheet that i have to scan into my computer and send to my teacher. it is an online math class.

Answer 46

Is there any way that you can just give me the answer so that i can finish the worksheet?