Question 729539
.
Alison deposits $500 into a new savings account that earns 5 percent interest compounded annually. 
If Alison makes no additional deposits or withdrawals, how many years will it take for the amount 
in the account to double?
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<pre>
We start from the standard formula for the future value of a compounded account

    FV = {{{500*(1+0.05)^n}}}.


We write this equation for the doubled future value

    1000 = {{{500*1.05^n}}}.


Here 'n' is the unknown number of the years (= number of compounding) to find.


We simplify equation (1) step by step

    {{{1000/500}}} = {{{1.05^n}}}.

    2 = {{{1.05^n}}}.


Take logarithm of both sides 

    log(2) = n*log(1.05).


Express and calculate 'n'

    n = {{{log((2))/log((1.05))}}} = 14.21  (approximately).


The number of compounding is an integer number - so, we must round this decimal 14.21
to the closest GREATER integer 15 in order for the bank be in position to make the last compounding.


<U>ANSWER</U>.  First time the compounded amount will exceed the doubled principal in 15 years.
</pre>

Solved.


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The solution/answer in the post by @lynnlo both are incorrect,
so, ignore his post for the peace in your mind.



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<H3>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;I N T E R E S T I N G </H3>


I checked today, on December &nbsp;15, &nbsp;2025, &nbsp;how &nbsp;Artificial &nbsp;Intelligence, &nbsp;Google &nbsp;AI &nbsp;Overview 
treats this problem.


It produces the same answer &nbsp;14.2 &nbsp;years as in the post by @lynnlo, without any critical analysis,
i.e., &nbsp;treats the problem &nbsp;INCORRECTLY.



I copied all references from the Google AI Overview output they are copy-pasted below.



[1] https://www.gauthmath.com/solution/1806082773471301/How-long-will-it-take-money-to-double-itself-if-invested-at-5-compounded-annuall

[2] https://askfilo.com/user-question-answers-smart-solutions/7-determine-how-long-will-it-take-money-to-double-itself-if-3331353538313435

[3] https://www.nerdwallet.com/banking/calculators/compound-interest-calculator

[4] https://study.com/skill/learn/finding-the-final-amount-in-a-word-problem-on-compound-interest-explanation.html

[5] https://www.vaia.com/en-us/textbooks/math/excursions-in-modern-mathematics-8-edition/chapter-10/problem-27-find-the-apr-of-a-bond-that-doubles-its-value-in-/

[6] https://askfilo.com/user-question-answers-smart-solutions/at-what-rate-a-sum-of-money-will-be-doubled-in-20-years-3133353733363230

[7] https://www.vaia.com/en-us/textbooks/math/contemporary-precalculus-5-edition/chapter-5/problem-75-you-have-5-grams-of-carbon-14-whose-half-life-is-/

[8] https://brainly.in/question/9793176

[9] https://www.math.uni.edu/~campbell/mdm/cont.html

[10] https://www.comerica.com/insights/wealth-management/wealth-preservation/financially-fit-clients-do-math.html

[11] https://www.quora.com/How-long-will-it-take-to-double-your-savings-if-you-earn-5-percent-interest-compounded-annually

[12] https://prepp.in/question/at-what-percent-of-compound-interest-per-annum-a-s-65e1ec82d5a684356ea12319



I checked each and every of these references.


Part of them produced the treatment for precisely this posed problem, but all 
produced treatments/answers were incorrect from the point of view of final rounding.


Other references were IRRELEVANT to the posed problem, which were especially sad to see.



Naturally, &nbsp;I informed &nbsp;Google &nbsp;AI &nbsp;Overview about this their mistake via their feedback system.


 . . . . . . . . . . . . . . . . . . . . 



In opposite, &nbsp;another &nbsp;Artificial &nbsp;Intelligence,  https://math-gpt.org/, &nbsp;treats the problem &nbsp;IDEALLY &nbsp;(precisely as in my solution).