Difference between revisions of "1984 AIME Problems/Problem 1"
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== Problem == | == Problem == | ||
| − | Find the value of <math>\displaystyle a_2+a_4+a_6+a_8+\ldots+a_{98}</math> if <math>\displaystyle a_1</math>, <math>\displaystyle a_2</math>, <math>\displaystyle a_3\ldots</math> is an arithmetic progression with common difference 1, and <math>\displaystyle a_1+a_2+a_3+\ldots+a_{98}=137</math>. | + | Find the value of <math>\displaystyle a_2+a_4+a_6+a_8+\ldots+a_{98}</math> if <math>\displaystyle a_1</math>, <math>\displaystyle a_2</math>, <math>\displaystyle a_3\ldots</math> is an [[arithmetic progression]] with common difference 1, and <math>\displaystyle a_1+a_2+a_3+\ldots+a_{98}=137</math>. |
== Solution == | == Solution == | ||
| − | {{ | + | One approach to this problem is to apply the formula for the sum of an [[arithmetic series]] in order to find the value of <math>a_1</math>, then use that to calculate <math>a_2</math> and sum another arithmetic series to get our answer. |
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| + | A somewhat quicker method is to do the following: for each <math>n \geq 1</math>, we have <math>a_{2n - 1} = a_{2n} - 1</math>. We can substitute this into our given equation to get <math>(a_2 - 1) + a_2 + (a_4 - 1) + a_4 + \ldots + (a_{98} - 1) + a_{98} = 137</math>. The left-hand side of this equation is simply <math>2(a_2 + a_4 + \ldots + a_{98}) - 49</math>, so our desired value is <math>\frac{137 + 49}{2} = 093</math>. | ||
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== See also == | == See also == | ||
* [[1984 AIME Problems/Problem 2 | Next problem]] | * [[1984 AIME Problems/Problem 2 | Next problem]] | ||
* [[1984 AIME Problems]] | * [[1984 AIME Problems]] | ||
Revision as of 11:05, 24 January 2007
Problem
Find the value of 
if 
, 
, 
is an arithmetic progression with common difference 1, and 
.
Solution
One approach to this problem is to apply the formula for the sum of an arithmetic series in order to find the value of 
, then use that to calculate 
and sum another arithmetic series to get our answer.
A somewhat quicker method is to do the following: for each 
, we have 
. We can substitute this into our given equation to get 
. The left-hand side of this equation is simply 
, so our desired value is 
.
