SOLUTION: If Naomi decides that she will invest ​$4 comma 500 per year in a 4​% annuity for the first ten​ years, ​$9,000 for the next ten​ years, and ​$13,500 for the next tenâ€

The major idea of the solution is that the FUTURE VALUE of so organized saving plan is the sum of FUTURE VALUES of 3 (three) annuities.


First annuity is 30 years ordinary annuity with the annual deposits of $4500 at 4% APY.


The second annuity is 20 years ordinary annuity with the annual deposits of $4500 at 4% APY.


And the third annuity is 10 years ordinary annuity with the annual deposits of $4500 at 4% APY.


The first annuity starts "this" year (or, better to say, it starts "today" with the first deposit at the end of "this"year); 

the second annuity starts 10 years after; and the third annuity starts 20 years after.


You can EASILY understand why it is so, by splitting each $9000 deposits in two deposits by $4500 each: 

    - then you continue the first annuity with $4500 deposits every year from 11-th year till 20-th year,

    - and you starts the second annuity with $4500 deposits annually from the year 11-th till 20-th.



And then splitting each $13500 deposits in three deposits by $4500 each:

    - then you continue the first annuity with $4500 deposits every year from 21-th year till 30-th year inclusively,

    - then you continue the second annuity with $4500 deposits every year from 21-th year till 30-th year inclusively,

    - and you starts the third annuity with $4500 deposits annually from the year 21-th till 30-th inclusively.



Thus the total future value after 30 years will be


    FV = FV1 + FV2 + FV3,  where


         FV1 = future value of the 4% annuity with $4500 deposits during 30 years;

         FV2 = future value of the 4% annuity with $4500 deposits during 20 years;

         FV3 = future value of the 4% annuity with $4500 deposits during 10 years.


The rest is just a technique.


    FV1 =  =  = 252382.22;


    FV2 =                  = 134001.36


    FV3 =                  =  54027.75.


Now the answer is the sum   252382.22 + 134001.36 + 54027.75 = 440411.33 dollars.


ANSWER.  440411.33 dollars.