Any mathmagicians out there? I have two equations that don't appear equal but they are--can you show me how?!

Question

1/4(5^k - 1) + 5^k = 1/4(5^(k+1) - 1)

If you're having a hard time with the exponents and all that, I typed it in Word and uploaded a picture of it

http://i134.photobucket.com/albums/q103/thepresident022/equation.jpg

So at any rate, can anyone figure this out? My dad and I tried for like half an hour and finally gave up! We got it a few steps down and plugged in some sample numbers and proved that it worked, but we just couldn't get it to where one side was exactly the same equation as the other.

Anyone wanna' give it a shot?

How do you get from first step to second step:

5^k-1+4*5^k=5^(k+1)-1
5*5^k -1 = 5^k+1 -1

Answer 1

(1/4)5^k - (1/4) + 5^k
= (5/4)5^k - 1/4
= (1/4)5^(k+1) - (1/4)
then you have the last step


§

Answer 2

(1/4)5^k + (1)5^k - 1/4 = 1/4(5^k)*5 - 1/4 => (5/4)5^k = (5/4)5^k => 5^k=5^k

Answer 3

LHS = 1/4(5^k - 1) + 5^k
= 1/4(5^k - 1) + 4/4*5^k
= 1/4(5^k - 1 + 4*5^k)
= 1/4(1*5^k - 1 + 4*5^k)
= 1/4(5*5^k -1)
= 1/4(5^(k+1) - 1)
= RHS

Please note that you should set your proof out as LHS = ... = ... = RHS. Do not write down both sides and manipulate them to get something that is equal or you cna prove anything.
E.g. lets 'prove' the 1=0 using that approach.
1=0 ... subtract 1/2 from both sides
1/2 = -1/2 ... square both sides
1/4 = 1/4 ... as its equal then 1=0
Rubbish. Set out your proofs carefully. :)

Answer 4

Multiply out both sides and do some sneaky rearrangments.
You have 5^k + 1/4(5^k) -1/4 = 1/4 (5^k)(5) - 1/4
or 5/4 (5^k) -1/4 = 5/4 (5^k) -1/4

Answer 5

1/4(5^k-1) + 5^k = 1/4(5^(k+1) -1)
5^k -1 + 4(5^k) = 5^(k+1) -1 Multiple both sides by 4
5^k + 4(5^k) = 5^(k+1) Subtract 1 from each side
5(5^k) = 5^(k+1) Sum up the left side
5^(k+1) = 5^(k+1) Note that 5(5^k) = 5^(k+1)

Looks like it works to me.

Answer 6

expand the terms
5^k/4 -1/4 +5^k = 5^(k+1)/4-1/4
cancel out 1/4 on each side
5^k/4+5^k = 5^(k+1)/4

mutiply the second term on left side by 4/4
5^k/4+4*5^k/4 = 5^(k+1)/4
(5^k+4*5^k)/4 = 5^(k+1)/4

times both side by 4
5^k+4*5^k= 5^(k+1)

on the left side
(1+4)5^k = 5*5^k = 5^1*5^k = 5^(1+k)

Answer 7

1/4(5^k - 1) + 5^k

(5^k - 1)/4 + 4(5^k)/4 {combine the terms}

[1(5^k) - 1 + 4(5^k)]/4

[5(5^k) - 1/4] {1(5^k) + 4(5^k) = 5(5^k)

[5^(k + 1) - 1]/4

Answer 8

1/4(5^k - 1) + 5^k = 1/4(5^(k+1) - 1)
1/4(5^k - 1) + 4*(5^k)/4 = 1/4(5^(k+1) - 1)
(1/4)(5^k-1+4*5^k)=1/4(5^(k+1)-1)
5^k-1+4*5^k=5^(k+1)-1
5*5^k -1 = 5^k+1 -1
5*5^k=5^k+1

NOw, remember that 5^k means: multiply 5 by itself K times...
On the left side, we do that, THEN multiply itselft ANOTHER times... that's K+1 multiplications total.
That's the same as 5^(k+1)
So we can rewrite:
5^(k+1)=5^(k+1)
They're equal!

Answer 9

so first divide each side by 1/4 than you ahve
(5^k - 1) + 5^k = (5^(k + 1) - 1)
than you do distributive for the right side of the equation and get:
(5^k - 1) + 5^k = (5^k+1) - 1
you can take the parenthesis away from the left side because nothing is getting multiplied or divided so you have:
5^k - 1 + 5^k = (5^k + 1) - 1
and than you can add together the 2 "5^k" on the left:
5^(k*2) - 1 = (5^k + 1) - 1
and than you can add 1 to both sides because yea:
5^(k*2) = (5^k + 1)
and than wait wtf they shouldn't be equal o.O LMAO

Answer 10

We can start by multiplying by 4 and expanding, getting:
5^k + 4 (5^k) =? 5^(k+1)
Or, 5(5*k) =? 5^(k+1)
Which is obviously true.