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Consider $A_1,A_2,A_3,.....A_n$ and$B_1,B_2,B_3,.....B_n$ $\ge20$ are two different Arithmetic progression such that $\frac{A_n}{B_1}=\frac{\sum_{i=1}^{n}2A_i}{\sum_{i=1}^{n}B_i}=\frac{B_n}{A_1}=4$, then find the value of

i) $\frac{A_n}{B_n}$

ii)$\frac{B_{10}-B_8}{A_{12}-A_{11}}$

I am not able to proceed as so many ratios are there.

Samar Imam Zaidi
  • 8,800
  • You need to use the fact that you have arithmetic progressions. Let $d$ be the difference of the $A$ series and $e$ the difference of the $B$ series, so $A_n=A_1+(n-1)d, B_n=B_1+(n-1)e$. That gives you five unknowns, $A_1,B_1,d,e,n$. Everything is proportional to $A_1$, so you can set it to $1$. You have three equations in four unknowns and I don't see how to eliminate another one. Follow it through and some other combination of the variables may not matter. The first is $\frac {B_1}{A_1}$ and the second is $\frac {2e}d$ – Ross Millikan May 05 '18 at 14:52

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