Let b i 1 for i = 1 , 2 , … , 101 . Suppose log e b 1 , log e b 2 , … , log e b 101 are in Arithmetic Progression (A.P.) with common difference log e 2 . Suppose a 1 , a 2 , … , a 101 are in A.P. such that a 1 = b 1 and a 51 = b 51 . If t =

Question

Let $b_{i} > 1$ for $i = 1, 2, \dots, 101 .$ Suppose $\log_{e} b_{1}, \log_{e} b_{2}, \dots, \log_{e} b_{101}$ are in Arithmetic Progression (A.P.) with common difference $\log_{e} 2 .$ Suppose $a_{1}, a_{2}, \dots, a_{101}$ are in A.P. such that $a_{1} = b_{1}$ and $a_{51} = b_{51} .$ If $t = b_{1} + b_{2} + \dots + b_{51}$ and $s = a_{1} + a_{2} + \dots + a_{53}$ , then [2016]

Smart Achievers · Accepted Answer

(2)

$\log_{e} b_{1}, \log_{e} b_{2}, \dots, \log_{e} b_{101} are in A.P.$

$\Rightarrow b_{1}, b_{2}, \dots, b_{101} are in G.P.$

$Also a_{1}, a_{2}, \dots, a_{101} are in A.P., where a_{1} = b_{1}$ are $a_{51} = b_{51} .$

$∴$ $b_{2}, b_{3}, \dots, b_{50} are G.M.'s and a_{2}, a_{3}, \dots, a_{50} are A.M.'s between b_{1} and b_{51} .$

$∵$ $G.M. < A.M. \Rightarrow b_{2} < a_{2}, b_{3} < a_{3}, \dots, b_{50} < a_{50}$

$∵$ $b_{1} + b_{2} + \dots + b_{51} < a_{1} + a_{2} + \dots + a_{51}$

$\Rightarrow t < s$

$Also a_{1}, a_{51}, a_{101} are in A.P. and b_{1}, b_{51}, b_{101} are in G.P.$

$∵$ $a_{1} = b_{1} and a_{51} = b_{51}$ $∴$ $b_{101} > a_{101}$

Solution

1 $s > t and a_{101} > b_{101}$

2 $s > t and a_{101} < b_{101}$

3 $s < t and a_{101} > b_{101}$

4 $s < t and a_{101} < b_{101}$

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Solution

Q. Let bi>1 for i=1,2,…,101. Suppose logeb1,logeb2,…,logeb101 are in Arithmetic Progression (A.P.) with common difference loge2. Suppose a1,a2,…,a101 are in A.P. such that a1=b1 and a51=b51. If t=b1+b2+⋯+b51 and s=a1+a2+⋯+a53, then [2016]

1 s>t and a101>b101

2 s>t and a101<b101

3 s<t and a101>b101

4 s<t and a101<b101

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1 $s > t and a_{101} > b_{101}$

2 $s > t and a_{101} < b_{101}$

3 $s < t and a_{101} > b_{101}$

4 $s < t and a_{101} < b_{101}$