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Let An=(3/4)-(3/4)^2+(3/4)^3+….+(-1)^(n-...

Let `A_n=(3/4)-(3/4)^2+(3/4)^3+….+(-1)^(n-1)(3/4)^n and B_n = 1-A_n. find the least odd nastural numebr s`n_0, so that B_ngtA_n Aangen_0`

Text Solution

Verified by Experts

The correct Answer is:
6
`a_(n)=3/4-(3/4)^(3)+(3/4)^(3)+…+(-1)^(n-1)(3/4)^(n)`
`(((3)/(4)(1-(-3)/(4))^(n)))/(1-((3)/(4)))=(3)/(7)(1-((-3)/(4))^(n))`
Now, `b_(n)=1-a_(n) and b_(n)gta_(n)` for `ngen_(0)`
`therefore1-a_(n)gta_(n)` or `2a_(n)lt1`
or `6/7[1-(-3/4)?^(n)]lt1`
or `-(-3/4)^(n)lt1/6`
or `(-3)^(n+1)lt2^(2n-1)`
for n to be even, inequality always holds. For n to be odd, it holds for `nge7`. Therefore, the least natural number for which it holds is 6.
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