document.write( "Question 1137944: You want to be able to withdraw $20,000 from your account each year for 20 years after you retire.\r
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\n" ); document.write( "\n" ); document.write( "If your account earns 8% interest, how much will you need to deposit each year until retirement to achieve your retirement goals? \n" ); document.write( "
Algebra.Com's Answer #755822 by ikleyn(52786)\"\" \"About 
You can put this solution on YOUR website!
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\n" ); document.write( "\n" ); document.write( "Let me re-formulate the problem to make it precisely correct.\r
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document.write( "    You want to be able to withdraw $20,000 from your account at the beginning of each year for 20 years after you retire.\r\n" );
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document.write( "    You expect to retire in 15 years.\r\n" );
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document.write( "    If your account earns 8% interest compounded annually, how much will you need to deposit each year until retirement \r\n" );
document.write( "    to achieve your retirement goals?\r\n" );
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\n" ); document.write( "\n" ); document.write( "            I will solve this re-formulated problem in two steps.\r
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\n" ); document.write( "\n" ); document.write( "Step  1.\r
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document.write( "First, I will determine how much money X should be accumulated on the account during 15 years to the time  of retiring,\r\n" );
document.write( "in order for to have enough to withdraw $20000 at the beginning of each year for 20 years.\r\n" );
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document.write( "By withdrawing $20000 each year, my compound account (the remaining money) still earns 8% per annum,  so everything works as it was \r\n" );
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document.write( "considered/described at the lesson  Withdrawing a certain amount of money periodically from a compounded saving account  in this site.\r\n" );
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document.write( "Thus the formula for the account value before it starts recharging is\r\n" );
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document.write( "    M = \"20000%2A%28%28%281-%281%2B0.08%29%5E%28-20%29%29%29%2F0.08%29\",     (1)\r\n" );
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document.write( "which gives us the value at the account of\r\n" );
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document.write( "    M  = 196362.95 dollars.\r\n" );
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\n" ); document.write( "\n" ); document.write( "Step  2. \r
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document.write( "Now I am in position to determine how much I should deposit at the end of each year during 15 years to the starting moment of the retiring\r\n" );
document.write( "to accumulate  196362.95 dollars in my account. \r\n" );
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document.write( "It is the standard Annuity saving plan, and the formula is\r\n" );
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document.write( "    196362.95 = \"D%2A%28%28%281%2B0.08%29%5E15-1%29%2F0.08%29\",     (2)\r\n" );
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document.write( "where D is the annual deposit amount.\r\n" );
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document.write( "The multiplier  \"%28%281%2B0.08%29%5E15-1%29%2F0.08\" = 27.15211,\r\n" );
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document.write( "which implies from equation (2)  that  D = \"196362.95%2F27.15211\" = 7231.96.\r\n" );
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document.write( "It is your answer:  During 15 years, you should deposit $7231.96 at the end of each year to your account,\r\n" );
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document.write( "                    in order to have the income of $20000 annually from yours account for 20 years.\r\n" );
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\n" ); document.write( "\n" ); document.write( "My lessons in this site associated with annuity saving plans and retirement plans are \r
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\n" ); document.write( "\n" ); document.write( "    - Geometric progressions\r
\n" ); document.write( "\n" ); document.write( "    - The proofs of the formulas for geometric progressions \r
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\n" ); document.write( "\n" ); document.write( "    - Ordinary Annuity saving plans and geometric progressions\r
\n" ); document.write( "\n" ); document.write( "    - Annuity Due saving plans and geometric progressions\r
\n" ); document.write( "\n" ); document.write( "    - Solved problems on Ordinary Annuity saving plans \r
\n" ); document.write( "\n" ); document.write( "    - Withdrawing a certain amount of money periodically from a compounded saving account\r
\n" ); document.write( "\n" ); document.write( "    - Miscellaneous problems on retirement plans \r
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\n" ); document.write( "\n" ); document.write( "Also,  you have this free of charge online textbook in ALGEBRA-II in this site\r
\n" ); document.write( "\n" ); document.write( "    - ALGEBRA-II - YOUR ONLINE TEXTBOOK.\r
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\n" ); document.write( "\n" ); document.write( "The referred lessons are the part of this online textbook under the topic \"Geometric progressions\".\r
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\n" ); document.write( "\n" ); document.write( "Save the link to this textbook together with its description\r
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\n" ); document.write( "\n" ); document.write( "Free of charge online textbook in ALGEBRA-II
\n" ); document.write( "https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson\r
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\n" ); document.write( "\n" ); document.write( "into your archive and use when it is needed.\r
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\n" ); document.write( "\n" ); document.write( "Tutor @MathTherapy absolutely correctly noticed the typo in my formula, which led me to incorrect answer in my previous version.\r
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\n" ); document.write( "\n" ); document.write( "I just corrected and re-calculated it, getting the same answer as @MathTherapy.
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\n" ); document.write( "\n" ); document.write( "Dear tutor @MathTherapy, thanks for your notice !\r
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