Question

Suppose on December 31, 2000, a deposit of $100 is made in a savings account
that pays 10% annual interest. So one year after the
initial deposit, on December 31, 2001, the account will be credited with $10, and
have a value of $110. On December 31, 2002 that account will be credited with an
additional $11, and have value $121. Find a recursive relation that gives the value
of the account n years after the initial deposit.

Answer 1

i always find it helpful to write out a few "terms" of the sequence to see if a pattern develops

Answer 2

b1 = (100*.1) + 11

b2 = b1*.1 + 11
= ((100*.1 +11) *.1) + 11

etc

Answer 3

1.1 is a better fit

Answer 4

b3 = b2*1.1 + 11
= ((100*1.1 +11) *1.1 + 11)*1.1 + 11
= (100*1.1^2 + 11*1.1 + 11)* 1.1 + 11
= 100*1.1^3 + 11*1.1^2 + 11*1.1 + 11
= 100*1.1^3 + 11 (1.1^2 + 1.1 + 1)

so to follow a pattern emerging I would say that:

b4 = 100(1.1)^4 + 11(1.1^3 + 1.1^2 + 1.1 + 1)

b5 = 100(1.1)^5 + 11(1.1^4 + 1.1^3 + 1.1^2 + 1.1 + 1)

etc

Answer 5

since we are looking for the nth term, we need to make this more general

\[b_r = 100(1.1)^r + 11(1.1^{r-1} + 1.1^{r-2} + 1.1^{r-3}+...+1.1^3+1.1^2 + 1.1 + 1)\]

that last part is a geometric series

Answer 6

It looks right, does this satisfy the definition of recursiveness?

Answer 7

Being that the last part is a geometric series, would it be better to write it in sigma summation notation?

Answer 8

im thinking this is turning into explicit

the recurrsion is simply:
\[b_{n+1}=1.1(b_n)+11\]

Answer 9

where b0 = 100

Answer 10

much clearer, and recursive due to n calling itself

Answer 11

\[b_n = 100(1.1)^n + 11(\frac{1-1.1^{n-1}} {1-1.1})\]

might be the equation to determine the nth year explicitly

Answer 12

is this last definition considered recursive?

Answer 13

definition never was my strong point :)

i dont think it would be considered recursive, but it should get you the nth year by plugging in n=year.

the bn+1 = 1.1(bn)+11; b0=100 is recursive since you cycle thru it n times

but a google search would prolly help this to be more definitive

Answer 14

I'm satisfied with the recursive definition, and laying out some of the initial steps makes it very evident...thank you!

Answer 15

youre welcome, and good luck ;)