2018 UNM-PNM Statewide High School Mathematics Contest II Problems/Problem 1

Problem

Let $x \ne y$ be two real numbers. Let $x,a_1,a_2,a_3,y$ and $b_1,x,b_2,b_3,y,b_4$ be two arithmetic sequences.

Calculate $\frac{b_4-b_3}{a_2-a_1}$.

Solution

Let $d_1$ be the common difference in the first sequence and $d_2$ the common difference in the second sequence. Thus, $b_4=x+4d_2$ and $b_3=x+2d_2$. In addition, $a_2=x+2d_1$ and $a_1=x+d_1$. Consequently \[\frac{b_4-b_3}{a_2-a_1}=\frac{x+4d_2-x-2d_2}{x+2d_1-x-d_1}\] or \[\frac{b_4-b_3}{a_2-a_1}=\frac{2d_1}{d_2}\] Since $d_2=\frac{y-x}{3}$ and $d_1=\frac{y-x}{4}$, we have \[\frac{2d_1}{d_2}=\frac{\frac{2y-2x}{3}}{\frac{y-x}{4}}\] which simplifies to $\boxed{\frac83}$. - Juno

See also

2018 UNM-PNM Contest II (ProblemsAnswer KeyResources)
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First question
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Problem 2
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All UNM-PNM Problems and Solutions
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