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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!! :sweat:

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!! :sweat:

= 3(2^k)+3(2(2^k))+3(2^k)+1

= 12(2^k) +1 ??

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!! :sweat:

hmm, i dun really understand how u get the line in bold

which is it? the two different questions you have posted.. are very different.

induction has a very unique format, if the first one posted is about induction too, then it's a very different story.

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

so i need to simplify the LHS close to the RHS form

ok.. to simplify everything, ill just type the actual question out... btw, thx again for all those helping or have helped me

Using induction, prove that (n^3) + 2 < 2^n for all n > 9

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 (n^3) + 2 be f(n)

since proof is for n > 9

let n be 10,

f(10) = 1002 < 2^10 = 1024

since proof is for n > 9

let n be 10,

f(10) = 1002 < 2^10 = 1024

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)

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?

but induction is a very weird topic - sometimes you are asked to do it because it is well, part of the syllabus. and if the question asked you to use induction.. then...? :dunno:

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

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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