Question 1091741: Please help.
The population of town was 525,000 in 1992 and is growing annually at the rate of 1.75%. Write a recursive sequence Pn for the population. State the first term P1 for the sequence.
I tried to solve...
It is geometric sequence, I think.
So, An=A n-1*r (n & n-1 is smaller letter)
Pn=P n-1 *(0.0175) This is recursive sequence for Pn.
Then how do I state the P1?
Thank you.
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! P1 is equal to 525,000
r is equal to 1.75% / 100 which is equal to .0175.
geometric sequence formula is Pn = P1 * r ^ (n-1)
if you follow that strict interpretation, you would think that Pn = P1 * .0175 ^ (n-1).
unfortunately that would be wrong.
r in this case is what you multiply 525,000 by in order to get the next larger term in the sequence.
that means you have to make r equal to 1 + r, since the next larger number is 525,000 + r * 525,000 which becomes 525,000 * (1 + r).
your common ratio, in this case, becomes 1 + r which is equal to 1.0175.
now the formula becomes Pn = 525,000 * 1.0175 ^ (n-1)
P1 is equal to 525,000.
common ratio of r is equal to 1.0175
n is equal to the number of terms in the sequence.
while this is a correct formula for a geometric sequence, it is not a recursive formula.
a recursive formula builds on the results of the previous iteration.
therefore, the recursive formula should be Pn+1 = Pn * 1.0175.
it can also be stated as Pn = Pn-1 * 1.0175.
what it's saying is that the current term is equal to the previous term * 1.0175.
your P1 would be 525,000
P2 would be P1 * 1.0175
P3 would be P2 * 1.0175
etc.
the geometric series formula and the recursive formula are related in the following manner.
geometric series formula would say that P3 = P1 * 1.075 ^ 2
recursive formula says that P3 = P2 * 1.0175
but P2 is equal to P1 * 1.075
therefore P3 is equal to P1 * 1.075 * 1.075
this makes P3 equal to P1 * 1.075 ^ 2
that's exactly what the geometric series formula is saying.
what's the difference?
with the geometric series formula, you don't need to know Pn-1 to find Pn.
you only need to know P1 and the common ratio.
with the recursive formula, you need to know Pn-1 in order to find Pn.
hope this helps.
let me know if you need anything further regarding this problem.
there can also be confusion between common ratio and interest rate.
it's true that 1.75% yields an interest rate of .0175.
that's an interest rate, but it's not a common ratio.
the common ratio, in this case becomes 1 + the interest rate which makes the common ratio equal to 1.0175.
different formulas use different definitions of r.
r can be interpreted as interest rate which gets you .0175.
r can also be interpreted as common ratio which gets you 1.0175 in this problem.
same letter but different interpretation.
your common ratio in this case is 1 plus the interest rate.
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