Question

Find the final amount in the following retirement​ account, in which the rate of return on the account and the regular contribution change over time. $289 per month invested at 4​%, compounded​ monthly, for 5 years; then $431 per month invested at 5​%, compounded​ monthly, for 5 years. What is the amount in the account after 10 years?

Answer 1

Solution:
	Amount in the account after 10 years	53,900.29	in 2 decimal
		53,900	in whole number
Notes:	You have not specified the decimal in which final answer is required I have given two , if question is silent then go with 1st decimal or go as required in question.
Working Notes:
	To get Amount in the account after 10 years, we have to compute future value of annuity at end of 5th year of first annuity and then that value is compounded for next 5 years at new rate of interest. And second annuity also we compute future value of annuity for next 5 years and added to the total value of 1st annuity value we get at end of 10 th year to Know Amount in the account after 10 years.
1st annuity of first 5 years
	Future value of annuity = P x ((1+i)^n - 1)/i
	P= Monthly deposits = $289 per month
	i= 4%/12
	n= no. Of payments x no. Of years
	= 5 x 12 =60
	Future value of annuity at end of 5th year=??
	Future value of annuity at end of 5th year
	= P x ((1+i)^n - 1)/i
	= 289 x ((1+(4%/12))^60 - 1)/(4%/12)
	=$19,160.4046947820
Now	This value will be compounded at new rate of interest for next 5 years .
	Future value at end of 10 year = Present value at end of 5th year x (1+new rate )^period
	Where
	Future value at end of 10 year of 1s annuity = ??
	Present value at end of 5th year = Future value of annuity at end of 5th year of 1st annuity = deposit we are doing at end of 5th year = $19,160.4046947820
	new rate = rate per month =(5%/12)
	period = No of years x no of months = 5 x 12 =60
	Future value at end of 10 year of 1st annuity
	= Present value at end of 5th year x (1+new rate )^period
	= $19,160.4046947820 x (1+(5%/12))^60
	= $19,160.4046947820 x (1+(5%/12))^60
	=24,589.67165
At last
2nd annuity of next 5 years
	Future value of annuity = P x ((1+i)^n - 1)/i
	P= Monthly deposits = $431 per month
	i= 5%/12
	n= no. Of payments x no. Of years
	= 5 x 12 =60
	Future value of annuity at end of 5th year=??
	Future value of 2nd annuity at end of next 5th year
	= P x ((1+i)^n - 1)/i
	= 431 x ((1+(5%/12))^60 - 1)/(5%/12)
	=$29,310.6217
Amount in the account after 10 years
=Future value of 2nd annuity at end of next 5th year + Future value at end of 10 year of 1st annuity
=$29,310.6217 + 24,589.67165
=$53,900.29335
Please feel free to ask if anything about above solution in comment section of the question.