Difference between revisions of "2011 AMC 10A Problems/Problem 4"
(→Solution) |
(→Solution 3) |
||
| (27 intermediate revisions by 11 users not shown) | |||
| Line 1: | Line 1: | ||
| + | ==Problem== | ||
Let X and Y be the following sums of arithmetic sequences: | Let X and Y be the following sums of arithmetic sequences: | ||
| − | <cmath> \begin{eqnarray*}X &=& 10+12+14+\cdots+100,\\ Y &=& 12+14+16+\cdots+102.\end | + | <cmath> \begin{eqnarray*}X &=& 10+12+14+\cdots+100,\\ Y &=& 12+14+16+\cdots+102.\end{eqnarray*} </cmath> |
| − | What is the value of Y - X? | + | What is the value of <math>Y - X? </math> |
<math> \textbf{(A)}\ 92\qquad\textbf{(B)}\ 98\qquad\textbf{(C)}\ 100\qquad\textbf{(D)}\ 102\qquad\textbf{(E)}\ 112 </math> | <math> \textbf{(A)}\ 92\qquad\textbf{(B)}\ 98\qquad\textbf{(C)}\ 100\qquad\textbf{(D)}\ 102\qquad\textbf{(E)}\ 112 </math> | ||
| − | ==Solution== | + | ==Solution 1== |
We see that both sequences have equal numbers of terms, so reformat the sequence to look like: | We see that both sequences have equal numbers of terms, so reformat the sequence to look like: | ||
| Line 14: | Line 15: | ||
X = 10 \ + \ &12 + 14 + \cdots + 100\\ | X = 10 \ + \ &12 + 14 + \cdots + 100\\ | ||
\end{align*}</cmath> | \end{align*}</cmath> | ||
| − | From here it is obvious that Y - X = 102 - 10 = <math>\boxed{92 \ \ | + | From here it is obvious that <math>Y - X = 102 - 10 = \boxed{92 \ \mathbf{(A)}}</math>. |
| + | |||
| + | ===Note=== | ||
| + | Another way to see this is to let the sum <math>12+14+16+...+100=x.</math> So, the sequences become | ||
| + | <cmath>\begin{align*} | ||
| + | X = 10+x \\ | ||
| + | Y= x+102 \\ | ||
| + | \end{align*}</cmath> | ||
| + | |||
| + | Like before, the difference between the two sequences is <math>Y-X=102-12=92.</math> | ||
| + | |||
| + | ==Solution 2== | ||
| + | |||
| + | We see that every number in Y's sequence is two more than every corresponding number in X's sequence. Since there are 46 numbers in each sequence, the difference must be | ||
| + | <math>46\cdot 2=\boxed{92}</math> | ||
| + | |||
| + | ==Solution 3== | ||
| + | |||
| + | <cmath>\begin{align*} | ||
| + | X&=10+12+14+\cdots +100 \\ | ||
| + | Y&=X-10+102 = X+92 \\ | ||
| + | Y-X &= (X+92)-X \\ | ||
| + | &= \boxed{92} \quad \quad \textbf{(A)}\\ | ||
| + | \end{align*} </cmath> | ||
| + | |||
| + | |||
| + | - <math>\text{herobrine-india}</math> | ||
| + | |||
| + | ==Solution 4== | ||
| + | |||
| + | In an actual contest, this would probably take too much time but is nevertheless a solution. | ||
| + | The general formula for computing sums of any arithmetic sequence where <math>x</math> is the number of terms, <math>f</math> is the first term and <math>l</math> is the last term is <math>\frac{(f+l)x}{2}</math>. If one uses that formula for both sequences, they will get <math>2530</math> as the sum for <math>X</math> and <math>2622</math> as the sum for <math>Y</math>. | ||
| + | Subtracting <math>X</math> from <math>Y</math>, one will get the answer <math>\boxed{92 \text{\textbf{ (A)}}}</math>. | ||
| + | - danfan | ||
| + | |||
| + | ==Video Solution== | ||
| + | https://youtu.be/L6utIF9FzPQ | ||
| + | |||
| + | ~savannahsolver | ||
== See Also == | == See Also == | ||
{{AMC10 box|year=2011|ab=A|num-b=3|num-a=5}} | {{AMC10 box|year=2011|ab=A|num-b=3|num-a=5}} | ||
| + | {{MAA Notice}} | ||
Latest revision as of 10:02, 17 June 2024
Contents
Problem
Let X and Y be the following sums of arithmetic sequences:
What is the value of 
Solution 1
We see that both sequences have equal numbers of terms, so reformat the sequence to look like:
From here it is obvious that 
.
Note
Another way to see this is to let the sum 
So, the sequences become
Like before, the difference between the two sequences is 
Solution 2
We see that every number in Y's sequence is two more than every corresponding number in X's sequence. Since there are 46 numbers in each sequence, the difference must be
Solution 3
- 
Solution 4
In an actual contest, this would probably take too much time but is nevertheless a solution.
The general formula for computing sums of any arithmetic sequence where 
is the number of terms, 
is the first term and 
is the last term is 
. If one uses that formula for both sequences, they will get 
as the sum for 
and 
as the sum for 
.
Subtracting from , one will get the answer 
.
- danfan
Video Solution
~savannahsolver
See Also
| 2011 AMC 10A (Problems • Answer Key • Resources) | ||
| Preceded by Problem 3 |
Followed by Problem 5 | |
| 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 | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
