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## Finding your annuity, Finding your reverse annuity

~TLDR at the bottom~
1. The problem statement, all variables and given/known data

The Big question:
 Suppose you can put $250/month into a retirement account that pays a 6% (annualized) return on investment. When you reach 65 years of age, how much will you have in your retirement account? [“Annualized” refers to “on a yearly basis” —this is why you see things like “i/12” in formulas (or “i/n” in general). 6% annually would correspond to 6%/12 = 0.5% per month.] At this point when you retire, you move your money into a low-risk account that pays 3% (annualized) return on investment. If you plan to live for 15 more years, how much can you draw out each month to live on? Variables Account 1: Code: P=250 R=.005 N=12 T=46 Variables Account 2: Code: A1=155123.83 R=.0025 N=12 T=15 2. Relevant equations Annuity Formula(Given): Code: A=P[(1+(r/n))nt)-1] ______________ (r/n) 3. The attempt at a solution For the savings account it was easy. My Attempt: Code: A=250((1+(.005/12))^((12)(46))-1)/(.005/12) A=$155123.83
(This answer was received using my age of 19)

After this point it shows my attempts at solving the second part of the question.
If you know how to solve it skip this next very long and very tedious section.
If you are trying to solve it and am not sure how,
the next part may inspire you or just confuse you.
_______________________________________________________________________ ___
The second part is tricky. I must find the compound interest while removing money.
My First Attempt:
Solve for P using annuity
Variables:
Code:
A=155123.83
R=.0025
T=15
N=12
Formula:
Code:
155123.83=P[(1+(.0025/12))^((12)(15))-1]
______________
(.0025/12)
P=$12912.18 This cannot be true if A is the annuity than P would be what you add per month to get A. _______________________________________________________________________ _____ My Seccond Attempt: Sum of a Geometric Sequence Variables: Code: A1=155123.83 R=1.0025 N=180 Formula: Code: ((155123.83)(1-1.0025^180)) ____________________ (1-1.0025) I felt this would give me the end amount but without the money removed each month which is unhelpful :(. _______________________________________________________________________ __ My Third Attempt: Indicated Sum Variables: Code: X=180 I=155123.83 F=(i-x)(1-1.0025^180))/(1-1.0025) Assuming: X $\Sigma$ F I Formula: Code: 180 Ʃ (i-x)(1.0025) 155123.83 Which i see no way i would be abel to solve for x this was also a dead end :( __________________________________________________________________ My Final Attempt: Recursion Variables: Code: A1=155123.83 AN=An-1-x)(1.0025) N>=180 Formula: Code: AN=An-1-x)(1.0025) A1=155123.83 A2=(155123.83-x)(*1.0025) But as you see i cannot solve for x without knowing what A180 equals. and i cannot find what A180 equals without X. _______________________________________________________________________ ____ I'm sure there is a way to do it that i am just missing. As you can see i did try.... I do not necessarily need the answer just a push in the right direction. And if you get the answer show your work so i do not lose my mind wondering. _______________________________________________________________________ ____ TLDR: Answer the problem and show work ;) TLDR EXTENDED: $250/month into a retirement account that pays a 6% (annualized) return on investment. When I am 65 the amount in the account will be A. A=\$155123.83
 you put the money from the first account into another account. this account pays a 3% (annualized) return on investment. At age 65 If you plan to live for 15 more years, how much can you draw out each month to live on?
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