Sequences and series (1 Viewer)

[Fortian09]

Omgosh i cant believe i'm asking this but i need some help on sequences and series

1. On the 1^st January, 1957, a person joins a superannuation fund by investing $3000 at 9%p.a. compound interest. A similar amount is invested at the beginning of each subsequent year until the person retires on 31^st Decenberm 1984.
a) Show that the accumulated value of the investment at the date of retirement is $369406 correct to the nearest dollar.
b)

 

[AlexJB]

If you get a question identical to this in the HSC, or just the future you could probably just sub the values in to the formula we find at the end. It's better to get an understanding of how to reach the final answer though.

There may probably be a better way of doing this. I never sat down and actually had a teacher tell me exactly what to do. Learnt by myself, but this way works good for me. Someone else can post another way if they want.

Basically, I just broke it down as to how much he would have in his account at the start and end of each year. You are usually able to recognise a pattern and hence write a formula for each question if you do it this way.

Good luck.

 

[Fortian09]

Hmm you know for the limiting sums for sequences and series...

What happens if |r|>1? instead of it being -1<r<1?
Because i have that problem in my Geom.Seq. qns for limiting sums

 

[lyounamu]

Hmm you know for the limiting sums for sequences and series...


What happens if |r|>1? instead of it being -1<R<1?< p> Because i have that problem in my Geom.Seq. qns for limiting sums

eh? care to explain?

for limiting sum, r is always between -1 and 1.

 

[shaon0]

Hmm you know for the limiting sums for sequences and series...

What happens if |r|>1? instead of it being -1<r<1?
Because i have that problem in my Geom.Seq. qns for limiting sums

It wouldn't really be a limiting sum then because the series will go to infinity.

 

[annabackwards]

It wouldn't really be a limiting sum then because the series will go to infinity.

Definitely, it wouldn't be a limiting sum at all because |r| is NOT <1

 

[cutemouse]

Omgosh i cant believe i'm asking this but i need some help on sequences and series

1. On the 1st January, 1957, a person joins a superannuation fund by investing $3000 at 9%p.a. compound interest. A similar amount is invested at the beginning of each subsequent year until the person retires on 31st December 1984.
a) Show that the accumulated value of the investment at the date of retirement is $369406 correct to the nearest dollar.

n=28 years
r=9%

The 1st $3000 accumulates to 3000(1.09)²⁸
The 2nd $3000 accumulates to 3000(1.09)²⁷
The 3rd $3000 accumulates to 3000(1.09)²⁶
...and so on...
The 28th $3000 accumulates to 3000(1.09)¹

Total investment = 3000(1.09)²⁸+3000(1.09)²⁷+...+3000(1.09)¹

Reversing order and factorising by 3000:
Total investment = 3000[1.09+1.09²+1.09³+...1.09²⁸]

As G.P. exists in parenthesis...

S_n=[a(rⁿ-1)]/(r-1)
a=1.09
r=1.09
n=28

Just sub those values in, and then times the whole thing by 3000, and you should get that answer, unless I've made some stupid error.