Difference between revisions of "1984 AIME Problems/Problem 1"
Math Kirby (talk | contribs) m (→Solution 2) |
(→Solution) |
||
Line 24: | Line 24: | ||
<math> \frac{137 - 97(49) + 2(49)^2}{2} = \fbox{093} </math>. | <math> \frac{137 - 97(49) + 2(49)^2}{2} = \fbox{093} </math>. | ||
+ | |||
+ | === Solution 3 === | ||
+ | A better approach to this problem is to notice that from <math>a_{1}+a_{2}+\cdots a_{98}=137</math> that each element with an odd subscript is 1 from each element with an even subscript. Thus, we note that the sum of the odd elements must be <math>\frac{137-49}{2}</math>. Thus, if we want to find the sum of all of the even elements we simply add <math>49</math> common differences to this giving us <math>\frac{137-49}{2}+49=93</math>. | ||
== See also == | == See also == |
Revision as of 19:07, 23 December 2013
Problem
Find the value of if
,
,
is an arithmetic progression with common difference 1, and
.
Solution
Solution 1
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
.
Solution 2
If is the first term, then
can be rewritten as:
Our desired value is so this is:
which is . So, from the first equation, we know
. So, the final answer is:
.
Solution 3
A better approach to this problem is to notice that from that each element with an odd subscript is 1 from each element with an even subscript. Thus, we note that the sum of the odd elements must be
. Thus, if we want to find the sum of all of the even elements we simply add
common differences to this giving us
.
See also
1984 AIME (Problems • Answer Key • Resources) | ||
Preceded by First Question |
Followed by Problem 2 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |