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If an=3/4-(3/4)^2+(3/4)^3+...(-1)^(n-1)(...

If `a_n=3/4-(3/4)^2+(3/4)^3+...(-1)^(n-1)(3/4)^n` and `b_n=1-a_n`, then find the minimum natural number n, such that `b_n>a_n`

Text Solution

Verified by Experts

The correct Answer is:
7
`B_(n) = 1 - A_(n) gt A_(n)`
`rArr A_(n) lt (1)/(2) rArr (3)/(4) ((1 - (-(3)/(4))^(n)))/(1 + (3)/(4)) lt (1)/(2)`
`rArr (-(3)/(4))^(n) gt - (1)/(6)`
Obviously, it is true for all even values of n.
But for
`n = 1, -(3)/(4) lt - (1)/(6)`
`n = 3, (-(3)/(4))^(3) = - (27)/(64) lt - (1)/(6)`
`n = 5, (-(3)/(4))^(5) = - (243)/(1024) lt - (1)/(6)`
and for `n = 7`,
`(-(3)/(4))^(7) = - (2187)/(12288) gt - (1)/(6)`
Hence, minium odd natural number `n_(0) = 7`
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