Difference between revisions of "1984 AIME Problems/Problem 1"
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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. | 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. | ||
− | 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} = \boxed{ | + | 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} = \boxed{93}</math>. |
== Solution 2 == | == Solution 2 == |
Latest revision as of 15:33, 24 July 2024
Contents
Problem
Find the value of if
,
,
is an arithmetic progression with common difference 1, and
.
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
.
Or, since the sum of the odd elements is 44, then the sum of the even terms must be .
Solution 4
We want to find the value of , which can be rewritten as
.
We can split
into two parts:
and
Note that each term in the second expression is
greater than the corresponding term, so, letting the first equation be equal to
, we get
. Calculating
by sheer multiplication is not difficult, but you can also do
. We want to find the value of
. Since
, we find
.
.
- PhunsukhWangdu
Solution 5
Since we are dealing with an arithmetic sequence,
We can also figure out that
Thus,
~kempwood
Video Solution by OmegaLearn
https://youtu.be/re8DTg-Lbu0?t=234
~ pi_is_3.14
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 |