2011 AMC 10A Problems/Problem 4

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 \\      &= \boxed{92} \quad \quad \textbf{(A)}\\ \end{align*}


- $\text{herobrine-india}$

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 $\frac{(f+l)x}{2}$. 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 $\boxed{92 \text{\textbf{ (A)}}}$. - danfan

Video Solution

https://youtu.be/L6utIF9FzPQ

~savannahsolver

See Also

2011 AMC 10A (ProblemsAnswer KeyResources)
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. AMC logo.png