We were completing some classwork for Annuities and finding the Sum of a G.P for simple situations as well as finding the time take for an annuity to reach a given value.
Given the context we wanted to see, if you start saving $10,000 a year how long would you have to save to purchase a house and we modelled the growth of the value of the house.
A house is worth $150,000 and appreciates at 7.5% P.A compounded annually and the Future Value can be found with:
$150000(1.075)^n$
The annuity can be represented by the equations $135000(1.08^n - 1)$
For what value of n are they equal? We have tried solving making the equations equal to each other and solve.
We have got as far as:
$150000(1.075)^{n} = 1235000(1.08^n -1)$
$10/9 (1.075)^{n} = 1.08^{n} -1$
We can find the solution of approximately 36 by trial and error and graphing on Desmos but we are looking for an algebraic solution.
We have been doing logs and Sum of GP and we thought we could solve simultaneously but we're stuck at the above.
We tried substituting let a = 1.075 and b = 1.08
giving:
$10a^n - 9b^{n} = -9$
but the different powers of n has us stuck. We think we could be able to change the base of the powers???
Sorry for first post
