Difference between revisions of "2011 AMC 10A Problems/Problem 4"

Revision as of 11:03, 19 October 2023

Problem

Let X and Y be the following sums of arithmetic sequences:

$\begin{eqnarray*}X &=& 10+12+14+\cdots+100,\\ Y &=& 12+14+16+\cdots+102.\end{eqnarray*}$

What is the value of $Y - X?$

$\textbf{(A)}\ 92\qquad\textbf{(B)}\ 98\qquad\textbf{(C)}\ 100\qquad\textbf{(D)}\ 102\qquad\textbf{(E)}\ 112$

Solution 1

We see that both sequences have equal numbers of terms, so reformat the sequence to look like:

$\begin{align*} Y = \ &12 + 14 + \cdots + 100 + 102\\ X = 10 \ + \ &12 + 14 + \cdots + 100\\ \end{align*}$ From here it is obvious that $Y - X = 102 - 10 = \boxed{92 \ \mathbf{(A)}}$ .

Note

Another way to see this is to let the sum $12+14+16+...+100=x.$ So, the sequences become $\begin{align*} X = 10+x \\ Y= x+102 \\ \end{align*}$

Like before, the difference between the two sequences is $Y-X=102-12=92.$

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: $46\cdot 2=\boxed{92}$

Solution 3

$\begin{align*} X&=10+12+14+\cdots +100 \\ Y&=X-10+102 = X+92 \\ Y-X &= (X+92)-X \\ Y-X &= X-X+92 \\ Y-X &= 0+92 \\ Y-X &= \boxed{92} \quad \quad \textbf{(A)}\\ \end{align*}$ $\blacksquare$

- $\text{herobrine-india}$

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 $x$ is the number of terms, $f$ is the first term and l is the last term is $((f+l)/2)x$ . If one uses that formula for both sequences, they will get $2530$ as the sum for $X$ and $2622$ as the sum for $Y$ . Subtracting $X$ from $Y$ , one will get the answer \boxed{$92 \text{\textbf{ (A)}}}$ (Error compiling LaTeX. Unknown error_msg). - danfan

Video Solution

https://youtu.be/L6utIF9FzPQ

~savannahsolver

@@ Line 49: / Line 49: @@
 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 l is the last term is <math>((f+l)/2)x</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>92 \text{\textbf{ (A)}}</math>.
+Subtracting <math>X</math> from <math>Y</math>, one will get the answer \boxed{<math>92 \text{\textbf{ (A)}}}</math>.
 - danfan

2011 AMC 10A (Problems • Answer Key • Resources)
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