# Difference between revisions of "2021 JMPSC Invitationals Problems/Problem 4"

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+ | ==Solution 2== | ||

+ | Add both equations to get <math>2(x_{n+1})=5x_n-y_n</math>, and subtract both equations to get <math>2(y_{n+1})=5y_n-x_n</math>, so now we bash: <math>x_1=7</math> and <math>y_1=1</math>. <math>x_2=17</math> and <math>y_2=-1</math>. <math>x_3=43</math> and <math>y_3=-11</math>. <math>x_4=113</math> and <math>y_4=-49</math>, <math>x_5=\frac{614}{2}=\boxed{307}</math> | ||

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+ | ~Geometry285 | ||

==See also== | ==See also== | ||

− | #[[2021 JMPSC | + | #[[2021 JMPSC Invitationals Problems|Other 2021 JMPSC Invitationals Problems]] |

− | #[[2021 JMPSC | + | #[[2021 JMPSC Invitationals Answer Key|2021 JMPSC Invitationals Answer Key]] |

#[[JMPSC Problems and Solutions|All JMPSC Problems and Solutions]] | #[[JMPSC Problems and Solutions|All JMPSC Problems and Solutions]] | ||

{{JMPSC Notice}} | {{JMPSC Notice}} |

## Latest revision as of 21:08, 11 July 2021

## Contents

## Problem

Let and be sequences of real numbers such that , , and, for all positive integers ,

Find .

## Solution

We notice that Since we are given that and , we can plug these values in to get that

Similarly, we conclude that

Adding and gives us Dividing both sides by yields

~mahaler

## Solution 2

Add both equations to get , and subtract both equations to get , so now we bash: and . and . and . and ,

~Geometry285

## See also

- Other 2021 JMPSC Invitationals Problems
- 2021 JMPSC Invitationals Answer Key
- All JMPSC Problems and Solutions

The problems on this page are copyrighted by the Junior Mathematicians' Problem Solving Competition.