So has everyone done their half yearlys?? (1 Viewer)
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~*HSC 4 life*~
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Yeah i had my ext 1 today, totally failed it 
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
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
George W. Bush
Banned
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.
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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cows_go_moo
Throbbing Member
i feel so bad, i had my ext exam on monday and if i get everything i did correct i can get 80%. but thats not likely
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
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
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
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
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>
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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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>)
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>)
C
CaR
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yep...my ext1 maths exam was the day before my 2U exam
recieved my marks yesterday...i passed...-__-''
recieved my marks yesterday...i passed...-__-''
mazza_728
Manda xoxo
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!
When do we learn this telescoping series thing? and wat topic is it under?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.
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.