Difference between revisions of "AoPS Wiki talk:Problem of the Day/June 15, 2011"
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==Solution== | ==Solution== | ||
− | {{ | + | ===Solution 1=== |
+ | We can solve this problem by a bit of trial and error. | ||
+ | |||
+ | We can guess she rode <math>5</math> days and we get <math>7+10+13+16+19=(13)(5)=65</math> since the mean is clearly <math>13</math> and there are <math>5</math> terms. | ||
+ | |||
+ | That's a bit too small. | ||
+ | |||
+ | We can add <math>22</math> to <math>65</math> and get <math>87</math>. That's still too small. | ||
+ | |||
+ | Now, we add <math>25</math> to get <math>112</math>, which is the total we want. | ||
+ | |||
+ | We started with <math>5</math> days and added <math>2</math> more, so there are <math>7</math> days in total. We can add <math>7-1=6</math> days to her starting day, Monday, to find her final day. Our answer is thus <math>\boxed{Sunday}</math> | ||
+ | |||
+ | ===Solution 2=== | ||
+ | On the first day, Jenny rode <math>7</math> miles. On the second day, she rode <math>7+3=10</math> miles. On the third day, she rode <math>10+3=13</math> miles. | ||
+ | |||
+ | This is the sequence <math>7,10,13,...</math> which is an arithmetic sequence: first term <math>7</math>, common difference <math>3</math>. | ||
+ | |||
+ | We are trying to find the number of terms <math>n</math> such that the <math>n\text{th}</math> partial sum of the sequence is <math>112</math>. | ||
+ | |||
+ | The formula for the sum of a partial sequence is <math>\frac{n}{2}[2a+(n-1)d]</math>, where <math>a</math> is the first term, <math>n</math> is the number of terms, and <math>d</math> is the common difference. (Try to derive it!) | ||
+ | |||
+ | Let <math>a=7</math> and <math>d=3.</math> Then we have: | ||
+ | |||
+ | <math>\frac{n}{2}[14+3(n-1)]=112</math> | ||
+ | |||
+ | <math>n[14+3(n-1)]=224</math> | ||
+ | |||
+ | <math>14n+3n(n-1)=224</math> | ||
+ | |||
+ | <math>14n+3n^2-3n=224</math> | ||
+ | |||
+ | <math>3n^2+11n-224=0</math> | ||
+ | |||
+ | <math>(n-7)(3n+32)=0</math> | ||
+ | |||
+ | The second root is not an integer, so the workout lasted for <math>n=7</math> days. The <math>7\text{th}</math> day after Monday is <math>\boxed{\text{Sunday}}</math>. |
Latest revision as of 11:55, 15 June 2011
Contents
Problem
AoPSWiki:Problem of the Day/June 15, 2011
Solution
Solution 1
We can solve this problem by a bit of trial and error.
We can guess she rode days and we get
since the mean is clearly
and there are
terms.
That's a bit too small.
We can add to
and get
. That's still too small.
Now, we add to get
, which is the total we want.
We started with days and added
more, so there are
days in total. We can add
days to her starting day, Monday, to find her final day. Our answer is thus
Solution 2
On the first day, Jenny rode miles. On the second day, she rode
miles. On the third day, she rode
miles.
This is the sequence which is an arithmetic sequence: first term
, common difference
.
We are trying to find the number of terms such that the
partial sum of the sequence is
.
The formula for the sum of a partial sequence is , where
is the first term,
is the number of terms, and
is the common difference. (Try to derive it!)
Let and
Then we have:
The second root is not an integer, so the workout lasted for days. The
day after Monday is
.