So has everyone done their half yearlys?? (1 Viewer)

TooF

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Yeah, just did it today.. wasn't too bad.. i don't think :S Your half yearlies are really late, but its coolies coz u get the hols to study instead of cramming till wee hours of the morning :D
 

untamedanimal

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I just finished mine like 50 mins ago. It was out of 65 and im predicting around 55-60. We got this really hard question for induction (not really hard but harder than normal). See if anyone can do it. Im pretty sure i proved it.

A fibonacci sequence defines:
T(1) = 1, T(2) = 2 ... ,T(n) = T(n-1) + T(n-2) for n>2

Prove that T(1) + T(2) + ..... +T(n) = T(n+2) - 2

It stumped me till right at the end
 
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Test n=1 etc.
Assume true for n=k
i.e. t1 + t2 + t3.........+tk = t(k+2) - 2
Hence prove for n=k+1
LHS = t(k+2) - 2 + t(k+1) (skipping some steps ) RHS = t(k+3) -2
LHS = t(k+3) - 2 as T(n) = T(n-1) + T(n-2)
therefore LHS = RHS
principle of maths induction etc. etc.

I always thought fibonacci numbers went 1, 1, 2, 3, 5, 8 etc. ? ah well.
 
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Xayma

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Originally posted by George W. Bush
I always thought fibonacci numbers went 1, 1, 2, 3, 5, 8 etc. ? ah well.
They do, T(0) also equals 1
 

CM_Tutor

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A Fibonacci sequence is one with two initial terms, where every other term is the sum of the preceeding two. The most commonly discussed one is the one GWB and Xayma are mentioning, ie:

1, 1, 2, 3, 5, 8, 13, ...

And the next most famous is the one in this question, ie:

1, 2, 3, 5, 8, 13, ...

Another two would be:

12, 6, 18, 24, 42, 66, ...

and

-7, 3, -4, -1, -5, -6, -11, ...

All have the property that:

T<sub>1</sub> + T<sub>2</sub> + T<sub>3</sub> + ... + T<sub>n</sub> = T<sub>n+2</sub> - T<sub>2</sub>

of which the above is one example
 

Affinity

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try this:
T(1) + T(2) + ... + T(n)
+T(2) + T(3) + ... + T(n+1)
= T(3) + T(4) + ... + T(n+2)

(since T(1) + T(2) = T(3), T(2) + T(3) = T(4) etc.)
therefore

T(1) + T(2) + ... + T(n) = T(n+2) - T(2) = T(n+2) - 2
 

CM_Tutor

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Nice Proof :)

Extn 2 students, take note: Affinity's proof is a variation on the use of telescoping series, which appear in Extn 2 from time to time.

In this case, we have T<sub>n</sub> = T<sub>n+2</sub> - T<sub>n+1</sub>

So, T<sub>1</sub> + T<sub>2</sub> + T<sub>3</sub> + ... + T<sub>n</sub> = (T<sub>3</sub> - T<sub>2</sub>) + (T<sub>4</sub> - T<sub>3</sub>) + (T<sub>5</sub> - T<sub>4</sub>) + ... + (T<sub>n+2</sub> - T<sub>n+1</sub>) = T<sub>n+2</sub> - T<sub>2</sub>
 
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CM_Tutor

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Another question that could be asked would be about the relationship between these various sequences / series.

For example, consider the sequence defined by T<sub>1</sub> = 1, T<sub>2</sub> = 2, T<sub>n</sub> = T<sub>n-1</sub> + T<sub>n-2</sub>, for n > 2 - ie. the sequence in untamedanimal's question and the 'regular' Fibonacci sequence, F<sub>1</sub> = 1, F<sub>2</sub> = 1, F<sub>n</sub> = F<sub>n-1</sub> + F<sub>n-2</sub>, for n > 2.

Prove that T<sub>n</sub> = T<sub>1</sub> * F<sub>n-2</sub> + T<sub>2</sub> * F<sub>n-1</sub>, for n => 3

(Note: This is also a general property of all such sequences T<sub>n</sub>)
 
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CaR

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yep...my ext1 maths exam was the day before my 2U exam

recieved my marks yesterday...i passed...-__-''
 

mazza_728

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I had mine the other day got my results -- 59% im not that happy... the other girl in my class got 66%. i made some really stupid mistakes and yeah i just feel stupid! i was expecting to fail but still! im aiming for uai 90+ and if im getting marks like this im gonna get so dissapointed!
 

nike33

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Originally posted by mazza_728
I had mine the other day got my results -- 59% im not that happy... the other girl in my class got 66%. i made some really stupid mistakes and yeah i just feel stupid! i was expecting to fail but still! im aiming for uai 90+ and if im getting marks like this im gonna get so dissapointed!
66% was the top mark? is that 3u or 4u? anyway can you please post the test? :)
 

DcM

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Originally posted by CM_Tutor
Nice Proof :)

Extn 2 students, take note: Affinity's proof is a variation on the use of telescoping series, which appear in Extn 2 from time to time.
When do we learn this telescoping series thing? and wat topic is it under?
 

CM_Tutor

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Originally posted by DcM
When do we learn this telescoping series thing? and wat topic is it under?
I can't pont to any specific topic - I guess I'd call it Harder 3u, as it's really another trick for using with sequences and series. It is a concept that appears from time to time, that you are supposed to pick up, somehow. I can point to some questions using it, if you'd like, but if you want to do that we should move this discussion to the Extn 2 forum.
 

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