Given an A.P. and a G.P. with positive terms, | vedclass

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(B) Let the first term be $a_1$ and the second term be $a_2$. Since terms are positive,$a_1 > 0$ and $a_2 > 0$.For the $A.P.$,the common difference $d

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$a_n  2$

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(B) Let the first term be $a_1$ and the second term be $a_2$. Since terms are positive,$a_1 > 0$ and $a_2 > 0$.For the $A.P.$,the common difference $d = a_2 - a_1$. Thus,$a_n = a_1 + (n - 1)(a_2 - a_1)$.For the $G.P.$,the common ratio $r = \frac{a_2}{a_1}$. Thus,$b_n = a_1 \left(\frac{a_2}{a_1}ight)^{n-1}$.For $n = 3$,$a_3 = a_1 + 2(a_2 - a_1) = 2a_2 - a_1$.For $n = 3$,$b_3 = a_1 \left(\frac{a_2}{a_1}ight)^2 = \frac{a_2^2}{a_1}$.Consider $b_3 - a_3 = \frac{a_2^2}{a_1} - (2a_2 - a_1) = \frac{a_2^2 - 2a_1a_2 + a_1^2}{a_1} = \frac{(a_2 - a_1)^2}{a_1}$.Since $a_1 > 0$ and $(a_2 - a_1)^2 \ge 0$,$b_3 - a_3 \ge 0$,implying $b_3 \ge a_3$. If $a_1 
eq a_2$,then $b_3 > a_3$. By induction or properties of convex functions,$b_n > a_n$ for all $n > 2$.

Given an $A.P.$ and a $G.P.$ with positive terms,where the first and second terms of both progressions are equal. If $a_n$ and $b_n$ are the $n^{\text{th}}$ terms of the $A.P.$ and $G.P.$ respectively,then:

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