Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "1984 AIME Problems/Problem 1"

m
m (→‎Solution 1)
 
(25 intermediate revisions by 14 users not shown)
Line 1: Line 1:
 
== 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>a_2+a_4+a_6+a_8+\ldots+a_{98}</math> if <math>a_1</math>, <math>a_2</math>, <math>a_3\ldots</math> is an [[arithmetic progression]] with common difference 1, and <math>a_1+a_2+a_3+\ldots+a_{98}=137</math>.
 +
 
 +
== Solution 1 ==
  
−
== 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.
 
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} = 093</math>.
+
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 ==
 +
 
 +
If <math> a_1 </math> is the first term, then <math> a_1+a_2+a_3 + \cdots + a_{98} = 137 </math> can be rewritten as:
 +
 
 +
<math> 98a_1 + 1+2+3+ \cdots + 97 = 137 </math> <math>\Leftrightarrow</math>
 +
<math> 98a_1 + \frac{97 \cdot 98}{2} = 137 </math>
 +
 
 +
Our desired value is <math> a_2+a_4+a_6+ \cdots + a_{98} </math> so this is:
 +
 
 +
<math> 49a_1 + 1+3+5+ \cdots + 97 </math>
 +
 
 +
which is <math> 49a_1+ 49^2 </math>. So, from the first equation, we know <math> 49a_1 = \frac{137}{2} - \frac{97 \cdot 49}{2} </math>. So, the final answer is:
 +
 
 +
<math> \frac{137 - 97(49) + 2(49)^2}{2} = \fbox{93} </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=\fbox{93}</math>.
 +
 
 +
Or, since the sum of the odd elements is 44, then the sum of the even terms must be <math>\fbox{93}</math>.
 +
 
 +
== Solution 4 ==
 +
We want to find the value of <math>a_2+a_4+a_6+a_8+\ldots+a_{98}</math>, which can be rewritten as <math>a_1+1+a_2+2+a_3+\ldots+a_{49}+49 \implies a_1+a_2+a_3+\ldots+a_{49}+\frac{49 \cdot 50}{2}</math>.
 +
We can split <math>a_1+a_2+a_3+\ldots+a_{98}</math> into two parts:
 +
<cmath>a_1+a_2+a_3+\ldots+a_{49}</cmath> and <cmath>a_{50}+a_{51}+a_{52}+\ldots+a_{98}</cmath>
 +
Note that each term in the second expression is <math>49</math> greater than the corresponding term, so, letting the first equation be equal to <math>x</math>, we get <math>a_1+a_2+a_3+\ldots+a_{98}=137=2x+49^2 \implies x=\frac{137-49^2}{2}</math>. Calculating <math>49^2</math> by sheer multiplication is not difficult, but you can also do <math>(50-1)(50-1)=2500-100+1=2401</math>. We want to find the value of <math>x+\frac{49 \cdot 50}{2}=x+49 \cdot 25=x+1225</math>. Since <math>x=\frac{137-2401}{2}</math>, we find <math>x=-1132</math>. <math>-1132+1225=\boxed{93}</math>.
 +
 
 +
- PhunsukhWangdu
 +
 
 +
== Solution 5 ==
 +
 
 +
Since we are dealing with an arithmetic sequence,
 +
<cmath>a_2+a_4+a_6+a_8+\ldots+a_{98} = 49a_{50}</cmath>
 +
We can also figure out that
 +
<cmath>a_1+a_2+a_3+\ldots+a_{98} = a_1 + 97a_{50} = 137</cmath>
 +
<cmath>a_1 = a_{50}-49 \Rightarrow 98a_{50}-49 = 137</cmath>
 +
Thus, <math>49a_{50} = \frac{137 + 49}{2} = \boxed{93}</math>
 +
 
 +
~kempwood
 +
 
 +
== Video Solution by OmegaLearn ==
 +
https://youtu.be/re8DTg-Lbu0?t=234
 +
 
 +
~ pi_is_3.14
  
 
== See also ==
 
== See also ==
−
* [[1984 AIME Problems/Problem 2 | Next problem]]
+
{{AIME box|year=1984|before=First Question|num-a=2}}
−
* [[1984 AIME Problems]]
+
* [[AIME Problems and Solutions]]
 +
* [[American Invitational Mathematics Examination]]
 +
* [[Mathematics competition resources]]
  
−
[[Category:Intermediate Algebra Problems]]
+
[[Category: Intermediate Algebra Problems]]

Latest revision as of 14:33, 24 July 2024

Problem

Find the value of $a_2+a_4+a_6+a_8+\ldots+a_{98}$ if $a_1$, $a_2$, $a_3\ldots$ is an arithmetic progression with common difference 1, and $a_1+a_2+a_3+\ldots+a_{98}=137$.

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 $a_1$, then use that to calculate $a_2$ and sum another arithmetic series to get our answer.

A somewhat quicker method is to do the following: for each $n \geq 1$, we have $a_{2n - 1} = a_{2n} - 1$. We can substitute this into our given equation to get $(a_2 - 1) + a_2 + (a_4 - 1) + a_4 + \ldots + (a_{98} - 1) + a_{98} = 137$. The left-hand side of this equation is simply $2(a_2 + a_4 + \ldots + a_{98}) - 49$, so our desired value is $\frac{137 + 49}{2} = \boxed{93}$.

Solution 2

If $a_1$ is the first term, then $a_1+a_2+a_3 + \cdots + a_{98} = 137$ can be rewritten as:

$98a_1 + 1+2+3+ \cdots + 97 = 137$ $\Leftrightarrow$ $98a_1 + \frac{97 \cdot 98}{2} = 137$

Our desired value is $a_2+a_4+a_6+ \cdots + a_{98}$ so this is:

$49a_1 + 1+3+5+ \cdots + 97$

which is $49a_1+ 49^2$. So, from the first equation, we know $49a_1 = \frac{137}{2} - \frac{97 \cdot 49}{2}$. So, the final answer is:

$\frac{137 - 97(49) + 2(49)^2}{2} = \fbox{93}$.

Solution 3

A better approach to this problem is to notice that from $a_{1}+a_{2}+\cdots a_{98}=137$ 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 $\frac{137-49}{2}$. Thus, if we want to find the sum of all of the even elements we simply add $49$ common differences to this giving us $\frac{137-49}{2}+49=\fbox{93}$.

Or, since the sum of the odd elements is 44, then the sum of the even terms must be $\fbox{93}$.

Solution 4

We want to find the value of $a_2+a_4+a_6+a_8+\ldots+a_{98}$, which can be rewritten as $a_1+1+a_2+2+a_3+\ldots+a_{49}+49 \implies a_1+a_2+a_3+\ldots+a_{49}+\frac{49 \cdot 50}{2}$. We can split $a_1+a_2+a_3+\ldots+a_{98}$ into two parts: \[a_1+a_2+a_3+\ldots+a_{49}\] and \[a_{50}+a_{51}+a_{52}+\ldots+a_{98}\] Note that each term in the second expression is $49$ greater than the corresponding term, so, letting the first equation be equal to $x$, we get $a_1+a_2+a_3+\ldots+a_{98}=137=2x+49^2 \implies x=\frac{137-49^2}{2}$. Calculating $49^2$ by sheer multiplication is not difficult, but you can also do $(50-1)(50-1)=2500-100+1=2401$. We want to find the value of $x+\frac{49 \cdot 50}{2}=x+49 \cdot 25=x+1225$. Since $x=\frac{137-2401}{2}$, we find $x=-1132$. $-1132+1225=\boxed{93}$.

- PhunsukhWangdu

Solution 5

Since we are dealing with an arithmetic sequence, \[a_2+a_4+a_6+a_8+\ldots+a_{98} = 49a_{50}\] We can also figure out that \[a_1+a_2+a_3+\ldots+a_{98} = a_1 + 97a_{50} = 137\] \[a_1 = a_{50}-49 \Rightarrow 98a_{50}-49 = 137\] Thus, $49a_{50} = \frac{137 + 49}{2} = \boxed{93}$

~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
Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=1984_AIME_Problems/Problem_1&oldid=223544"