If α and β are the roots of the equation 375x2−25x−2=0, then limn→∞n∑r=1αr+limn→∞n∑r=1βr is equal to :
A
7116
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B
112
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C
29358
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D
21346
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Solution
The correct option is B112 Given, α and β are the roots of the equation 375x2−25x−2=0 Therefore α+β=−(−25375)=(25375) and α⋅β=(−2375) limn→∞n∑r=1αr+limn→∞n∑r=1βr=limn→∞n∑r=1(αr+βr) =limn→∞(α1+β1)+(α2+β2)+(α3+β3)+⋯+(αn+βn) =(α+α2+α3+⋯∞)+(β+β2+β3+⋯∞ Here we have infinite G.P series therefore =α1−α+β1−β=α+β−2αβ1−(α+β)+αβ =25375+43751−25375−2375=29348=112