Question

how to write this seqeunce as a summation

financial mathematics

Answer 1

first year 1

end of first year (1+R)

2nd year (1+r) + (1+i)

end of second yr (1+R)^2 + (1+i)(1+R)

third year (1+i)(1+r)^2 +(1+i)^2(1+r) + (1+i)^2

end of third yr (1+r)^3 + (1+i)(1+r)^2 + (1+i)^2(1+r)

4th year (1+r)^3 + (1+r)^2(1+i) + (1+r)^(1+i)^2 + (1+i)^3

etc... till the tenth yr

Answer 2

@Michele_Laino

Answer 3

hey again

i have looked at this and i have seen a pattern, this is the enlarged version of what we did before. however, my thought was wrong

Answer 4

you can see that at the beginning of each year there is an addition of (1+i)^n-1 to the sequence right?

Answer 5

yes!

Answer 6

now excluding the addition of this (1+i) term for each new year,

at the end of each year i can write the summation as:

\[\sum_{k=0}^{k=n-1}(1+r)^{n-(k+1)}(1+i)^{k}\]

Answer 7

first year 1

end of first year (1+R)

2nd year (1+r)

end of second yr (1+R)^2 + (1+i)(1+R)

third year (1+i)(1+r)^2 +(1+i)^2(1+r)

end of third yr (1+r)^3 + (1+i)(1+r)^2 + (1+i)^2(1+r)

4th year (1+r)^3 + (1+r)^2(1+i) + (1+r)^(1+i)^2
etc... till the tenth yr

Answer 8

see how i have removed the (1+i) term from the start of each new year

Answer 9

just so the summation can represent all those terms

Answer 10

but i'm not sure where i'm going with this. it looks interesting but i can't seem to add on that extra term so that everything is hunky dory

Answer 11

you can see the indices of each pair add up to n-1

Answer 12

or n

Answer 13

depending if its the beggining or end of the year

Answer 14

:/

Answer 15

please wait I try to write your summation

Answer 16

: )

Answer 17

i guess you can represent 1 as (1+i)^0

Answer 18

that could be a help

Answer 19

I got this expression at the beginning of each year:
\[\Large \sum\limits_{k = 0}^{n - 1} {{{\left( {1 + r} \right)}^{n - \left( {k + 1} \right)}}{{\left( {1 + i} \right)}^k}} \]

Answer 20

yep! thats what i got!

Answer 21

please can you write the expression at the end of the fourth year?

Answer 22

sure

Answer 23

give me a second

Answer 24

(1+r)^4 + (1+r)^3(1+i) +(1+r)^2(1+i)^2 + (1+r)(1+i)^3

Answer 25

can you see the pattern?

Answer 26

I got this expression:
\[\Large \sum\limits_{k = 0}^{n - 1} {{{\left( {1 + r} \right)}^{n - k}}{{\left( {1 + i} \right)}^k}} \]

Answer 27

is this for everything?

Answer 28

that expression is at the end of each year

Answer 29

let me check

Answer 30

yep! thats good by me

Answer 31

would u just use both summations and don't worry about combining?

Answer 32

yes! I think so!

Answer 33

so all you have to do is be careful which summation you are using?

Answer 34

that's right!

Answer 35

ah you've clarified it for me so easily!
cheers again!

Answer 36

thanks! :):)

Answer 37

buuddyyy!!