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Arithmetic Progressions

What can we say if \(a, x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, b\) are in arithmetic progression?
\[\frac{b-a}{6}=d$$ $$x_{2}=a+\frac{b-a}{3}$$ $$x_{1}+x_{3}+x_{5}=3a+9d$$ $$=3(a+3d)$$ $$=3\left( a+3\left( \frac{b-a}{6} \right) \right)$$ $$=3\left( a+\frac{b-a}{2} \right)$$ $$=3\left( \frac{1}{2}a+\frac{1}{2}b \right)$$ $$=\frac{3}{2}(a+b)\]

\(a\)
\(x_{2}=ar\)
\(b=ar^2\)

\(x_{2}^2=a^2r^2\)
\(x_{2}^2=ab\)
\((\frac{2a+b}{3})^2=ab\)
\(\frac{4a^2+4ab+b^2}{9}=ab\)
\(4a^2+4ab+b^2=9ab\)
\(4a^2-5ab+b^2=0\)
\((b-a)(b-4a)=0\)
\(b \neq a\) so \(b-4a=0\) and \(b=4a\)

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