Hi guys, any mathematician can help me with algebras? i cant figure out....
i need to make the equation below into 2(2^k) <=== (as in 2 to the power of k, times 2)
k^3 + 3(k)^2 + 3k + 1
any kind souls can help me out pls? greatly appreciated!!
Hi guys, any mathematician can help me with algebras? i cant figure out....
i need to make the equation below into 2(2^k) <=== (as in 2 to the power of k, times 2)
k^3 + 3(k)^2 + 3k + 1
any kind souls can help me out pls? greatly appreciated!!
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"Photography is an austere and blazing poetry of the real" -Ansel Adams
its not really a maths module...
hahah im taking IT actually :P
what i meant was, i need to simplify that long equation to 2(2^k)
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Sorry tried, but unable to help.
Is there any example given? in similar form.
thx for the help im actually doing proofing thru induction
i need to proof that (k+1)^3 < 2^(k+1)
so i need to simplify the LHS close to the RHS form
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sorry, please disregard my post. probably wrong calculations...
Just saw your last post. it would be easier if you had posted that in the 1st place. Don't be presumptious in assuming the answer.
Last edited by blive; 4th May 2008 at 03:49 PM.
"Photography is an austere and blazing poetry of the real" -Ansel Adams
oh ic, sorry blive....
i thought i was on the right track...
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sorry, I tried to help. getting rusty, but what do I know? I am an engineer, and we only use calculator to do simple calculations too!
How about assuming that x = (k+1)? then your equation becomes x^3 <2^x ==>x^2 <2 ===> k+1 < 2 ^(1/2) ==> substitute into original equation??
"Photography is an austere and blazing poetry of the real" -Ansel Adams
hmmm... let me try.. haha maybe easier...
thx for suggestion!
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let (n^3) + 2 be f(n)
since proof is for n > 9
let n be 10,
f(10) = 1002 < 2^10 = 1024
when n = g (arbitiary value), where g>9
f(g) = g^3 + 2 < 2^g -------- result (1) -- you have to hold this to be true
now take n = g+1, where g>9
f(g+1) = (g+1)^3 + 2 = g^3 + 3g^2 + 3g + 3
from result (1),
2g^3 + 2 < 2*2^g = 2^(g+1) --- result (2)
taking the difference between f(g+1) and result (2),
g^3 - 3g^2 - 3g + 1 has to be greater or equal to 0 for f(g+1) < result (2) [basic algebra rearrangement]
this is true when g > 9
as g^3 - 3g^2 -3g + 1 = (g+1) [(g-2)^2 -3]
if the above is true, and statement is true when n = 10 > 9
then by mathematical induction, the statement is true
btw i did this quickly, and something's not right because if you follow the steps, it seems to suggest that the statement is true for values smaller than n>9 (which isn't true, if you check the results), maybe someone at the jc level who's still familiar with mathematical induction can help.
Let me try...
Thus f(10) is true.
Assume that f(k) is true, ie (k^3) + 2 < 2^k
To prove f(k+1) is true, ie (k+1)^3 +2 < 2^ (k+1)
From LHS,
(k+1)^3 + 2 = (k^3 + 3k^2 + 3k +1) +2
< 2^k + 3k^2 + 3k +1
= 2^k + k^3 [3/k + 3/ (k^2) + 1/ (k^3)]
Now, for n >9, 3/k + 3/(k^2) + 1/ (k^3) <1
Hence, 2^k + k^3 > 2^k + k^3 [......]
so (k+1)^3 +2 < 2^k + k^3
< 2^k + k^3 + 2
< 2^k + 2^k
< 2 ^ (k+1) (shown)
hmmm, very chiim lehh
anyway, ill try to understand this and thx a lot for the help
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paiseh.. posted 4 times. think due to server lag
Actually is it a necessity to prove by induction? Isn't it easier to sketch two graphs and show that the intersection is at around 9?
Proving by induction is one of the many ways in which to use formal logic to prove theorems. Not much use in real life I guess =p
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wrong, actually.
most of science is formed on induction. if syndrome x occurs in the past, and happens in the present, therefore we would expect it to extend into the future.
but not many people realise that.
consider things like even your color green. goodman, a philosopher along with hume (who came first), questioned the logic of induction - there is no real logic actually, when it comes to daily life, that much is true. there is not much logic in it. but.. well, it's used.
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