Difference between revisions of "2002 AMC 10B Problems/Problem 19"
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− | We should realize that the two equations are 100 terms apart, so by subtracting the two equations in a form like | + | We should realize that the two equations are 100 terms apart, so by subtracting the two equations in a form like... |
+ | <cmath>(a_{101} - a_1) + (a_{102} - a_2) +... + (a_{200} - a_{100}) = 200-100 = 100 </cmath> | ||
− | + | ...we get the value of the common difference of every hundred terms one hundred times. So we have to divide the answer by one hundred to get ... | |
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<math>\frac{100}{100} = 1 </math> | <math>\frac{100}{100} = 1 </math> | ||
− | + | ...the common difference of every hundred terms. Then we have to simply divide the answer by hundred again to find the common difference between one term, therefore... | |
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+ | <math>\frac{1}{100} =\boxed{(\textbf{C})\ 0.01}</math> | ||
== Solution 2 == | == Solution 2 == | ||
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Now, let the common difference be <math> d </math>. Notice that <math> a_2-a_1=d </math>, so we merely need to find <math> d </math> to get the answer. The formula for an arithmetic sum is | Now, let the common difference be <math> d </math>. Notice that <math> a_2-a_1=d </math>, so we merely need to find <math> d </math> to get the answer. The formula for an arithmetic sum is | ||
− | <math> \frac{n}{2}(2a_1+d(n-1)) </math>, | + | <math> \frac{n}{2}\left(2a_1+d(n-1)\right) </math>, |
where <math> a_1 </math> is the first term, <math> n </math> is the number of terms, and <math> d </math> is the common difference. Now we use this formula to find a closed form for the first given equation and the sum of the given equations. For the first equation, we have <math> n=100 </math>. Therefore, we have | where <math> a_1 </math> is the first term, <math> n </math> is the number of terms, and <math> d </math> is the common difference. Now we use this formula to find a closed form for the first given equation and the sum of the given equations. For the first equation, we have <math> n=100 </math>. Therefore, we have | ||
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Now we have a system of equations in terms of <math> a_1 </math> and <math> d </math>. | Now we have a system of equations in terms of <math> a_1 </math> and <math> d </math>. | ||
− | Subtracting '''(1)''' from '''(2)''' eliminates <math> a_1 </math>, yielding <math> 100d=1 </math>, and <math> d=a_2-a_1=\frac{1}{100}=\boxed{(\ | + | Subtracting '''(1)''' from '''(2)''' eliminates <math> a_1 </math>, yielding <math> 100d=1 </math>, and <math> d=a_2-a_1=\frac{1}{100}=\boxed{(\textbf{C})\ 0.01}</math>. |
== Solution 3 == | == Solution 3 == | ||
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− | Now express each a_n in terms of first term a_1 and common difference x between consecutive terms | + | Now express each <math>a_n</math> in terms of first term <math>a_1</math> and common difference <math>x</math> between consecutive terms |
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− | Simplifying and canceling a_1 and x terms gives | + | Simplifying and canceling <math>a_1</math> and <math>x</math> terms gives |
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− | <math>100x=1</math> | + | <math>100x=1</math> |
+ | |||
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+ | <math>x=0.01=\boxed{(\textbf{C})\ 0.01}</math> | ||
+ | |||
+ | == Video Solution by OmegaLearn == | ||
+ | https://youtu.be/tKsYSBdeVuw?t=4410 | ||
+ | ~ pi_is_3.14 | ||
− | + | ==Video Solution== | |
+ | https://www.youtube.com/watch?v=38p1OD_ATFE ~David | ||
==See Also== | ==See Also== |
Latest revision as of 10:21, 22 August 2024
Contents
Problem
Suppose that is an arithmetic sequence with
What is the value of
Solution 1
We should realize that the two equations are 100 terms apart, so by subtracting the two equations in a form like...
...we get the value of the common difference of every hundred terms one hundred times. So we have to divide the answer by one hundred to get ...
...the common difference of every hundred terms. Then we have to simply divide the answer by hundred again to find the common difference between one term, therefore...
Solution 2
Adding the two given equations together gives
.
Now, let the common difference be . Notice that
, so we merely need to find
to get the answer. The formula for an arithmetic sum is
,
where is the first term,
is the number of terms, and
is the common difference. Now we use this formula to find a closed form for the first given equation and the sum of the given equations. For the first equation, we have
. Therefore, we have
,
or
. *(1)
For the sum of the equations (shown at the beginning of the solution) we have , so
or
*(2)
Now we have a system of equations in terms of and
.
Subtracting (1) from (2) eliminates
, yielding
, and
.
Solution 3
Subtracting the 2 given equations yields
Now express each in terms of first term
and common difference
between consecutive terms
Simplifying and canceling and
terms gives
Video Solution by OmegaLearn
https://youtu.be/tKsYSBdeVuw?t=4410
~ pi_is_3.14
Video Solution
https://www.youtube.com/watch?v=38p1OD_ATFE ~David
See Also
2002 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 18 |
Followed by Problem 20 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AMC 10 Problems and Solutions |
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