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How can I maximize a and r in a geometric sequence given the sum


Aunt and Uncle's fuel oil tank dip stick problemHow can I maximize the area of a rectangle given a continuous range of values for the length and width?Find common ratio of a geometric sequence given the sumLooking for rounded corner plane curve with certain properties (SIDESTEPPED)How to maximize the profit for the given equationMaximize volume of hyperrectangle given fixed sum of componentsArithmetic series and sequence + geometric sequenceUnits in geometric calculus?Scalar Multiplication, why do I keep getting this bizarre result?How can I maximize the area of a rectangular base and semicircular top when perimeter is not given













0












$begingroup$


I'm trying to divide a line into 5 sections for an art project. The line is 65 units long and I want the sections to be a geometric sequence rounded to the nearest unit. The math side of me wants to find an exact solution first.



That is I want $a+ar+ar^2+a^3+ar^4=65$. How can I maximize $a$ and $r$?



I tried using the closed form $a(1-r^5)/(1-r)=65$ taking the derivative w.r.t. $r$ but ended up just getting $ar^4=13$, which clearly can't be the correct solution if I want all the other terms to be less than $ar^4$. I also tried google-ing but everything I see shows either $r$ or $a$ is given in the problem. Thanks.






calculus







share|cite|improve this question









$endgroup$











  • $begingroup$
    We need to maximize $ar$ or $a,r$ separately
    $endgroup$
    – lab bhattacharjee
    19 hours ago















0












$begingroup$


I'm trying to divide a line into 5 sections for an art project. The line is 65 units long and I want the sections to be a geometric sequence rounded to the nearest unit. The math side of me wants to find an exact solution first.



That is I want $a+ar+ar^2+a^3+ar^4=65$. How can I maximize $a$ and $r$?



I tried using the closed form $a(1-r^5)/(1-r)=65$ taking the derivative w.r.t. $r$ but ended up just getting $ar^4=13$, which clearly can't be the correct solution if I want all the other terms to be less than $ar^4$. I also tried google-ing but everything I see shows either $r$ or $a$ is given in the problem. Thanks.






calculus







share|cite|improve this question









$endgroup$











  • $begingroup$
    We need to maximize $ar$ or $a,r$ separately
    $endgroup$
    – lab bhattacharjee
    19 hours ago













0












0








0


1



$begingroup$


I'm trying to divide a line into 5 sections for an art project. The line is 65 units long and I want the sections to be a geometric sequence rounded to the nearest unit. The math side of me wants to find an exact solution first.



That is I want $a+ar+ar^2+a^3+ar^4=65$. How can I maximize $a$ and $r$?



I tried using the closed form $a(1-r^5)/(1-r)=65$ taking the derivative w.r.t. $r$ but ended up just getting $ar^4=13$, which clearly can't be the correct solution if I want all the other terms to be less than $ar^4$. I also tried google-ing but everything I see shows either $r$ or $a$ is given in the problem. Thanks.






calculus







share|cite|improve this question









$endgroup$




I'm trying to divide a line into 5 sections for an art project. The line is 65 units long and I want the sections to be a geometric sequence rounded to the nearest unit. The math side of me wants to find an exact solution first.



That is I want $a+ar+ar^2+a^3+ar^4=65$. How can I maximize $a$ and $r$?



I tried using the closed form $a(1-r^5)/(1-r)=65$ taking the derivative w.r.t. $r$ but ended up just getting $ar^4=13$, which clearly can't be the correct solution if I want all the other terms to be less than $ar^4$. I also tried google-ing but everything I see shows either $r$ or $a$ is given in the problem. Thanks.






calculus




calculus






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked 20 hours ago









Jacob BreckenridgeJacob Breckenridge

11




11











  • $begingroup$
    We need to maximize $ar$ or $a,r$ separately
    $endgroup$
    – lab bhattacharjee
    19 hours ago
















  • $begingroup$
    We need to maximize $ar$ or $a,r$ separately
    $endgroup$
    – lab bhattacharjee
    19 hours ago















$begingroup$
We need to maximize $ar$ or $a,r$ separately
$endgroup$
– lab bhattacharjee
19 hours ago




$begingroup$
We need to maximize $ar$ or $a,r$ separately
$endgroup$
– lab bhattacharjee
19 hours ago










1 Answer
1






active

oldest

votes


















2












$begingroup$

You have one constraint, decide on what function of $a,r$ you want to maximize, use the constraint to solve for $a$ and plug in. For example, to maximize $ar$, note that
$$
a = frac65(1-r)1-r^5
$$
so you want to maximize
$$
ar = frac65r(1-r)1-r^5 = g(r),
$$
and now standard Calculus techniques apply, here is the solution from Wolfram Alpha.






share|cite|improve this answer









$endgroup$












  • $begingroup$
    Thanks, I was thinking it was some simple algebraic part I was missing and it turns out it was!
    $endgroup$
    – Jacob Breckenridge
    11 hours ago










Your Answer





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1 Answer
1






active

oldest

votes








1 Answer
1






active

oldest

votes









active

oldest

votes






active

oldest

votes









2












$begingroup$

You have one constraint, decide on what function of $a,r$ you want to maximize, use the constraint to solve for $a$ and plug in. For example, to maximize $ar$, note that
$$
a = frac65(1-r)1-r^5
$$
so you want to maximize
$$
ar = frac65r(1-r)1-r^5 = g(r),
$$
and now standard Calculus techniques apply, here is the solution from Wolfram Alpha.






share|cite|improve this answer









$endgroup$












  • $begingroup$
    Thanks, I was thinking it was some simple algebraic part I was missing and it turns out it was!
    $endgroup$
    – Jacob Breckenridge
    11 hours ago















2












$begingroup$

You have one constraint, decide on what function of $a,r$ you want to maximize, use the constraint to solve for $a$ and plug in. For example, to maximize $ar$, note that
$$
a = frac65(1-r)1-r^5
$$
so you want to maximize
$$
ar = frac65r(1-r)1-r^5 = g(r),
$$
and now standard Calculus techniques apply, here is the solution from Wolfram Alpha.






share|cite|improve this answer









$endgroup$












  • $begingroup$
    Thanks, I was thinking it was some simple algebraic part I was missing and it turns out it was!
    $endgroup$
    – Jacob Breckenridge
    11 hours ago













2












2








2





$begingroup$

You have one constraint, decide on what function of $a,r$ you want to maximize, use the constraint to solve for $a$ and plug in. For example, to maximize $ar$, note that
$$
a = frac65(1-r)1-r^5
$$
so you want to maximize
$$
ar = frac65r(1-r)1-r^5 = g(r),
$$
and now standard Calculus techniques apply, here is the solution from Wolfram Alpha.






share|cite|improve this answer









$endgroup$



You have one constraint, decide on what function of $a,r$ you want to maximize, use the constraint to solve for $a$ and plug in. For example, to maximize $ar$, note that
$$
a = frac65(1-r)1-r^5
$$
so you want to maximize
$$
ar = frac65r(1-r)1-r^5 = g(r),
$$
and now standard Calculus techniques apply, here is the solution from Wolfram Alpha.







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered 20 hours ago









gt6989bgt6989b

34.8k22456




34.8k22456











  • $begingroup$
    Thanks, I was thinking it was some simple algebraic part I was missing and it turns out it was!
    $endgroup$
    – Jacob Breckenridge
    11 hours ago
















  • $begingroup$
    Thanks, I was thinking it was some simple algebraic part I was missing and it turns out it was!
    $endgroup$
    – Jacob Breckenridge
    11 hours ago















$begingroup$
Thanks, I was thinking it was some simple algebraic part I was missing and it turns out it was!
$endgroup$
– Jacob Breckenridge
11 hours ago




$begingroup$
Thanks, I was thinking it was some simple algebraic part I was missing and it turns out it was!
$endgroup$
– Jacob Breckenridge
11 hours ago

















draft saved

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