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

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==Solution== | ==Solution== | ||

− | Since the sum of the first 2011 terms is 200, and the sum of the fist 4022 terms is 380, the sum of the second 2011 terms is 180. | + | Since the sum of the first <math>2011</math> terms is <math>200</math>, and the sum of the fist <math>4022</math> terms is <math>380</math>, the sum of the second <math>2011</math> terms is <math>180</math>. |

This is decreasing from the first 2011, so the common ratio (or whatever the term for what you multiply it by is) is less than one. | This is decreasing from the first 2011, so the common ratio (or whatever the term for what you multiply it by is) 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 9/ | + | Because it is a geometric sequence and the sum of the first 2011 terms is <math>200</math>, second <math>2011</math> is <math>180</math>, the ratio of the second <math>2011</math> terms to the first <math>2011</math> terms is <math>\frac{9}{10}</math>. Following the same pattern, the sum of the third <math>2011</math> terms is <math>\frac{9}{10}*180 = 162</math>. |

Thus, | Thus, | ||

− | 200+180+162=542 | + | <math>200+180+162=542</math> |

− | Sum of the first 6033 is <math>\framebox[1.3\width]{542.}</math> | + | Sum of the first <math>6033</math> is <math>\framebox[1.3\width]{542.}</math> |

## Revision as of 16:18, 13 July 2011

## 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 terms is , and the sum of the fist terms is , the sum of the second terms is . This is decreasing from the first 2011, so the common ratio (or whatever the term for what you multiply it by is) is less than one.

Because it is a geometric sequence and the sum of the first 2011 terms is , second is , the ratio of the second terms to the first terms is . Following the same pattern, the sum of the third terms is .

Thus,

Sum of the first is