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Turning a recursive formula into explicit?
A couple puts $750,000 into a retirement account paying 4.5% annual interest. They estimate that they will need to withdraw $75,000 a year to live on. Assume that on the last year interest is paid, and then the $75,000 is withdrawn.
Find a non-recursive formula for Sn, the amount of money left in the account after n years. First, find a pattern for S1, S2, S3, and S4. For example: S1 = 750,000(1.045) - 75000
The recursive formula is easy to find:
S(n) = S(n-1) * 1.045 - 75000
But how would I turn this into an explicit formula?
Sorry the recursive formula would be:
S(n) = S(n-1) * 1.045 - 75000; S(0) = 750,000
2 Answers
- fernando_007Lv 68 years agoFavorite Answer
From:
S(n) = S(n-1) * 1.045 - 75000 and
S(n-1) = S(n-2) * 1.045 - 75000 follows:
DS(n) = DS(n-1)*1.045 = DS(n-2)*1.045^2 = ... = DS(1)*1.045^(n-1),
where DS(k) = S(k) - S(k-1), and DS(1) = S(1)-S(0) = S(0)*0.045 - 75000.
On another hand: sum(from k=1 to n)[DS(k)] = (S(n)-S(n-1)) + (S(n-1)-S(n-2)) +...
+ (S(1)-S(0)) = S(n) - S(0), therefore:
S(n) = S(0) + sum(from k=1 to n)[DS(k)] = S(0) + DS(1)*sum(from k=1 to n) [1.045^(k-1)] =
= S(0) + DS(1)*(1.045^n - 1)/0.045 and finally:
S(n) = = S(0) + (S(0)*0.045 - 75000)*(1.045^n - 1)/0.045.
- Anonymous4 years ago
stressful to declare... yet probable no longer. Off the best of my head, i can't think of of any recursive sequences/series that cannot be made show, yet i'm optimistic i've got encountered it. this is lots greater durable to instruct that something can't be executed than to instruct something could be executed. after all, they got here across an show formula for Fibonacci.
