Induction proof question (1 Viewer)

[king.rafa]

I'm stuck on this question:
prove via induction:
1^4 + 2^4 + ... + (n-1)^4 + n^4 = 1/30 x n x (n+1) x (2n+1) x (3n^2 + 3n - 1)

Now I can do it one way but I'm not sure if it is allowed because it is sort of like cheating and its an unelegant proof. For example can you expand the RHS to one value, and then expand the LHS to the same value, and prove it that way?

It is a foolproof method, but will you still get full marks for that approach?

 

[Drongoski]

I'm stuck on this question:
prove via induction:
1^4 + 2^4 + ... + (n-1)^4 + n^4 = 1/30 x n x (n+1) x (2n+1) x (3n^2 + 3n - 1)

Now I can do it one way but I'm not sure if it is allowed because it is sort of like cheating and its an unelegant proof. For example can you expand the RHS to one value, and then expand the LHS to the same value, and prove it that way?

It is a foolproof method, but will you still get full marks for that approach?

Here's my attempted soln outline:

u can easily show it's true for n = 1

now assume true for n = k >= 1

i.e. 1^4 + 2^4 + . . . + k^4 = 1/30 x k(k+1)(2k+1)(3k^2 + 3k -1)

now 1^4 + . . . + k^4 + (k+1)^4 = 1/30 x k(k+1)(2k+1)(3k^2+3k-1) + (k+1)^4

This can be shown to be 1/30 x (k+1)[6k^4 + 39k^3 + 91k^2 + 89k + 30)]

The RHS for n = k+1 can also to be shown to be this expression

Therefore true for n = k+1

Therefore by induction true for all integer n >= 1


Edit

[hopefully such a question never set in an HSC exam; easy to get bogged down with algebra mistakes; because it is tedious (for me at least) to massage the expression for case of n = k+1 for RHS as 1/30 x (k+1)(k+2)(2(k+1)+1)(3(k+1)^2 + 3(k+1) - 1) I had to resort to the brute force approach].

 

[Drongoski]

Short answer: yes

I've had to do it the way you alluded to. Why don't u just post yr solution/method so we know exactly what u are talking about.

 

[GUSSSSSSSSSSSSS]

For example can you expand the RHS to one value, and then expand the LHS to the same value, and prove it that way?

how wuld u expand the LHS in that expression???
its an infinite series...

 

[Pwnage101]

how wuld u expand the LHS in that expression???
its an infinite series...

you dont need to...
plus its not an infinite series

 

[GUSSSSSSSSSSSSS]

you dont need to...
plus its not an infinite series

u dun have to?? she said that thats how she PLANNED to do it..

 

[Pwnage101]

i dont know what ur on about...

all u do is add (k+1)^4 to both sides, then u have the LHS u want - u dont need to do anything else here. All you need to do is show that

1/30 x k(k+1)(2k+1)(3k^2+3k-1) + (k+1)^4

is equal to

1/30 x (k+1)[6k^4 + 39k^3 + 91k^2 + 89k + 30)]

which is equal to

1/30 x (k+1) x (k+2) x (2(k+1)+1) x (3(k+1)^2 + 3(k+1) - 1)

which is all thats required...

EDIT: i think she meant expand the RHS you get from the induction hypothesis for (k+1) and show that its equal to the RHS of the induction hypothesis for k, plus (k+1)^4

as in all induction proofs, u dont need to expand the LHS....

 

[GUSSSSSSSSSSSSS]

i dont know what ur on about...

all u do is add (k+1)^4 to both sides, then u have the LHS u want - u dont need to do anything else here. All you need to do is show that

1/30 x k(k+1)(2k+1)(3k^2+3k-1) + (k+1)^4

is equal to

1/30 x (k+1)[6k^4 + 39k^3 + 91k^2 + 89k + 30)]

which is equal to

1/30 x (k+1) x (k+2) x (2(k+1)+1) x (3(k+1)^2 + 3(k+1) - 1)

which is all thats required...

EDIT: i think she meant expand the RHS you get from the induction hypothesis for (k+1) and show that its equal to the RHS of the induction hypothesis for k, plus (k+1)^4

as in all induction proofs, u dont need to expand the LHS....

OHHHHH HAHAHA i COMPLETELY misinterpreted her question then LOL

sorry