Loretta is $30$ years old and has wanted to be a millionaire by the time she is $50.$ Her

investments usually provide an annual rate of return of $12\%.$

a$)$ If she makes an initial investment of $$25,000,$ how much of a regular monthly payment should

she make so that after $20$ years she will accumulate $$1,000,000?$

b$)$ If she makes no initial investment, how much of a regular monthly payment should she make

so that after $20$ years she will accumulate $$1,000,000?$

c$)$ If she makes an initial investment of $$25,000,$ how much of a regular monthly payment should

she make so that after $30$ years she will accumulate $$1,000,000?$