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Difference between revisions of "2011 AIME II Problems/Problem 5"

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The sum of the first 2011 terms of the sequence is expressible as <math>a_1 + a_1r + a_1r^2 + a_1r^3</math> .... until <math>a_1r^{2010}</math>. The sum of the 2011 terms following the first 2011 is expressible as <math>a_1r^{2011} + a_1r^{2012} + a_1r^{2013}</math> .... until <math>a_1r^{4021}</math>. Notice that the latter sum of terms can be expressed as <math>(r^{2011})(a_1 + a_1r + a_1r^2 + a_1r^3...a_1r^{2010})</math>. We also know that the latter sum of terms can be obtained by subtracting 200 from 380, which then means that <math>r^{2011} = 9/10</math>. The terms from 4023 to 6033 can be expressed as <math>(r^{4022})(a_1 + a_1r + a_1r^2 + a_1r^3...a_1r^{2010})</math>, which is equivalent to <math>((9/10)^2)(200) = 162</math>. Adding 380 and 162 gives the answer of <math>\boxed{542}</math>.
 
The sum of the first 2011 terms of the sequence is expressible as <math>a_1 + a_1r + a_1r^2 + a_1r^3</math> .... until <math>a_1r^{2010}</math>. The sum of the 2011 terms following the first 2011 is expressible as <math>a_1r^{2011} + a_1r^{2012} + a_1r^{2013}</math> .... until <math>a_1r^{4021}</math>. Notice that the latter sum of terms can be expressed as <math>(r^{2011})(a_1 + a_1r + a_1r^2 + a_1r^3...a_1r^{2010})</math>. We also know that the latter sum of terms can be obtained by subtracting 200 from 380, which then means that <math>r^{2011} = 9/10</math>. The terms from 4023 to 6033 can be expressed as <math>(r^{4022})(a_1 + a_1r + a_1r^2 + a_1r^3...a_1r^{2010})</math>, which is equivalent to <math>((9/10)^2)(200) = 162</math>. Adding 380 and 162 gives the answer of <math>\boxed{542}</math>.
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==Solution 4==
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Let <math>S_n</math> be equal to the sum of the first <math>n</math> terms of the geometric sequence. <math>S_{2011} = 200</math> and <math>S_{4022} = 380</math>. Let <math>a</math> be the first term and <math>r</math> be the common difference. So <math>a(\frac{1-r^{2011}}{1-r}) = 200</math> and <math>a(\frac{1-r^{4022}}{1-r}) = 380</math>. We take the positive difference between the two equations. <math>a(\frac{r^{2011}-r^{4022}}{r-1}) = 180</math>. Now, we'll factor out <math>r^{2011}</math> so the equation becomes <math>ar^{2011}(\frac{1-r^{2011}}{r-1}) = 180</math>. Divide this equation by the first equation and we get <math>r^{2011} = 9/10</math>. We now just need to find the ratio of <math>S_{6033}</math> to <math>S_{2011}</math> multiplied by <math>S_{2011}</math> (It's easy to find the ratio because of common terms). <cmath> \frac{S_{6033}}{S_{2011}} = \frac{\frac{a(1-r^{6033})}{1-r}}{\frac{a(1-r^{2011})}{1-r}} = \frac{1-r^{6033}}{1-r^{2011}} = \frac{1-(\frac{9}{10})^{3}}{1-\frac{9}{10}} = \frac{\frac{271}{1000}}{\frac{1}{10}} = \frac{271}{100}</cmath>
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Now it's simple, we just need to multiply this value by <math>S_{2011}</math> (which is <math>200</math>) and we get our final answer of <math>\boxed{542}</math>.
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~ROGER8432V3
  
 
==Video Solution==
 
==Video Solution==

Latest revision as of 00:44, 1 September 2025

Problem

The sum of the first $2011$ terms of a geometric sequence is $200$. The sum of the first $4022$ terms is $380$. Find the sum of the first $6033$ terms.

Solution

Since the sum of the first $2011$ terms is $200$, and the sum of the first $4022$ terms is $380$, the sum of the second $2011$ terms is $180$. This is decreasing from the first 2011, so the common ratio is less than one.

Because it is a geometric sequence and the sum of the first 2011 terms is $200$, second $2011$ is $180$, the ratio of the second $2011$ terms to the first $2011$ terms is $\frac{9}{10}$. Following the same pattern, the sum of the third $2011$ terms is $\frac{9}{10}*180 = 162$.

Thus, $200+180+162=542$, so the sum of the first $6033$ terms is $\boxed{542}$.

Solution 2

Solution by e_power_pi_times_i

The sum of the first $2011$ terms can be written as $\dfrac{a_1(1-k^{2011})}{1-k}$, and the first $4022$ terms can be written as $\dfrac{a_1(1-k^{4022})}{1-k}$. Dividing these equations, we get $\dfrac{1-k^{2011}}{1-k^{4022}} = \dfrac{10}{19}$. Noticing that $k^{4022}$ is just the square of $k^{2011}$, we substitute $x = k^{2011}$, so $\dfrac{1}{x+1} = \dfrac{10}{19}$. That means that $k^{2011} = \dfrac{9}{10}$. Since the sum of the first $6033$ terms can be written as $\dfrac{a_1(1-k^{6033})}{1-k}$, dividing gives $\dfrac{1-k^{2011}}{1-k^{6033}}$. Since $k^{6033} = \dfrac{729}{1000}$, plugging all the values in gives $\boxed{542}$.

Solution 3

The sum of the first 2011 terms of the sequence is expressible as $a_1 + a_1r + a_1r^2 + a_1r^3$ .... until $a_1r^{2010}$. The sum of the 2011 terms following the first 2011 is expressible as $a_1r^{2011} + a_1r^{2012} + a_1r^{2013}$ .... until $a_1r^{4021}$. Notice that the latter sum of terms can be expressed as $(r^{2011})(a_1 + a_1r + a_1r^2 + a_1r^3...a_1r^{2010})$. We also know that the latter sum of terms can be obtained by subtracting 200 from 380, which then means that $r^{2011} = 9/10$. The terms from 4023 to 6033 can be expressed as $(r^{4022})(a_1 + a_1r + a_1r^2 + a_1r^3...a_1r^{2010})$, which is equivalent to $((9/10)^2)(200) = 162$. Adding 380 and 162 gives the answer of $\boxed{542}$.


Solution 4

Let $S_n$ be equal to the sum of the first $n$ terms of the geometric sequence. $S_{2011} = 200$ and $S_{4022} = 380$. Let $a$ be the first term and $r$ be the common difference. So $a(\frac{1-r^{2011}}{1-r}) = 200$ and $a(\frac{1-r^{4022}}{1-r}) = 380$. We take the positive difference between the two equations. $a(\frac{r^{2011}-r^{4022}}{r-1}) = 180$. Now, we'll factor out $r^{2011}$ so the equation becomes $ar^{2011}(\frac{1-r^{2011}}{r-1}) = 180$. Divide this equation by the first equation and we get $r^{2011} = 9/10$. We now just need to find the ratio of $S_{6033}$ to $S_{2011}$ multiplied by $S_{2011}$ (It's easy to find the ratio because of common terms). \[\frac{S_{6033}}{S_{2011}} = \frac{\frac{a(1-r^{6033})}{1-r}}{\frac{a(1-r^{2011})}{1-r}} = \frac{1-r^{6033}}{1-r^{2011}} = \frac{1-(\frac{9}{10})^{3}}{1-\frac{9}{10}} = \frac{\frac{271}{1000}}{\frac{1}{10}} = \frac{271}{100}\] Now it's simple, we just need to multiply this value by $S_{2011}$ (which is $200$) and we get our final answer of $\boxed{542}$.

~ROGER8432V3

Video Solution

https://www.youtube.com/watch?v=rpYphKOIKRs&t=186s ~anellipticcurveoverq

See also

2011 AIME II (Problems • Answer Key • Resources)
Preceded by
Problem 4 		Followed by
Problem 6
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All AIME Problems and Solutions

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