Harder 2Unit (1 Viewer)

[OLDMAN]

Yes, harder 2 unit.

Through the magic of compounding, capital C becomes
C(1+r)^n after n years. How much do we need to invest to be able to withdraw $1 at the end of year 1, 4 at the end of year 2, 9 at the end of year 3, 16 at the end of year 4, and so on in perpetuity?

Actuaries welcome, show the meds. you coulc also add!

 

[Affinity]

Yes, harder 2 unit.

senses some *sarcasm* in that sentence. more like q30 of the AMC.

by the way, would be interesting to generalize the question to finding the amount that is needed to be invested if at the end of n years n^i dollars is to be withdrawn, where i is an integer.

 

[OLDMAN]

Excellent!
You really know your series, particularly
1=1
4=1+3
9=1+3+5 and so on, or k^2=sum(j:1-->k)[2j-1]
For someone who may not have remembered this, the following alternative solution:

As Affinity, Investment=sum(1-->infinity)[k^2/(1+r)^k]
Substitute w=1/(1+r).
1+w+w^2+...=1/(1-w)
Differentiate and times by w, w/(1-w)^2=w+2w^2+3w^3+...
Differentiate and times by w again,
w(1+w)/(1-w)^3=(1^2)w+(2^2)w^2+(3^2)w^3+...
which yields the same answer as Affinity's.

 

[OLDMAN]

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by the way, would be interesting to generalize the question to finding the amount that is needed to be invested if at the end of n years n^i dollars is to be withdrawn, where i is an integer

Affinity
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Nice thought, may be difficult though.

 

[Affinity]

Differentiating! why didn't I think about that
guess I can't become a medic nor an actuary

 

[Affinity]

By the way, is it ok to perform these operations(differentiating, multiplying by a variable)on the infinite series in the HSC?

 

[Lazarus]

Originally posted by Affinity
By the way, is it ok to perform these operations(differentiating, multiplying by a variable)on the infinite series in the HSC?

If you come across one where it may be useful, certainly!

 

[turtle_2468]

generally, yes, unless there are a variable no. of terms in the series., e.g. the erroneous proof going:
x^2= x+x+x... +x (x terms RHS)
diff wrt x: 2x=1+1+1...+1
so 2x=x and x=0.
so... if the terms are functions of x, make sure the no. of terms isn't.
In HSC, probably not such a good idea _if_ you can find another way to do it. Just because, when you use dodgy ways to do things even in school exams, teachers can't be bothered figuring out what u did and just think you fudged it, so yeah. If possible do it by non-diff'd infinite methods, but then if it's nice, use it by all means!

 

[OLDMAN]

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If you come across one where it may be useful, certainly!
-Lazarus
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Thanks Lazarus. You're certainly allowed to when you differentiate/integrate both sides of a binomial expansion in 3-unit.

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senses some *sarcasm* in that sentence. more like q30 of the AMC.
-Affinity
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Didn't really mean to convey sarcasm, rather, meant to stress that tools to solve the problem could be obtained from 2-unit maths eg. geometric series, differentiating, compound interest.