Let An = \[\left( {{3 \over 4}} \right) - {\left( {{3 \over 4}} \right)^2} + {\left( {{3 \over 4}} \right)^3}\] -. . . . . + (-1)n-1 \[{\left( {{3 \over 4}} \right)^n}\] and Bn = 1 - An.
Then, the least dd natural numbr p, so that Bn > An , for all n\[ \ge \] p, is :
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Official solution
An = \[\left( {{3 \over 4}} \right) - {\left( {{3 \over 4}} \right)^2} + {\left( {{3 \over 4}} \right)^3} - .... + {\left( { - 1} \right)^{n - 1}}{\left( {{3 \over 4}} \right)^n}\]
Which in a G.P. with a = \[{{3 \over 4}}\], r = \[{{{ - 3} \over 4}}\] and number of terms = n
\[\therefore\,\,\,\] An = \[{{{3 \over 4}\left( {1 - {{\left( {{{ - 3} \over 4}} \right)}^n}} \right)} \over {1 - \left( {{{ - 3} \over 4}} \right)}} = {{{3 \over 4} \times \left( {1 - {{\left( {{{ - 3} \over 4}} \right)}^n}} \right)} \over {{7 \over 4}}}\]
\[ \Rightarrow \] An =73\[\left[ {1 - {{\left( {{{ - 3} \over 4}} \right)}\^n}} \right]\] \[\,\,\,\,\,\,\,\,\,\] . . . . . . . . . .(1)
As, Bn = 1 - An
For least odd natural number p, such that Bn > An
\[ \Rightarrow \] 1−An > An⇒1 > 2×An⇒An <21From eqn. (1), we get73\[ \times \] \[\left[ {1 - {{\left( {{{ - 3} \over 4}} \right)}^n}} \right] < {1 \over 2}\] \[ \Rightarrow \] 1 - \[{\left( {{{ - 3} \over 4}} \right)^n} < {7 \over 6}\]
\[ \Rightarrow \] 1−67<(4−3)n⇒6−1<(4−3)nAs n is odd, then(4−3)n=−43nSo6−1<−(43)n⇒61>(43)nlog\left( {{1 \over 6}} \right) ( = n log\]\left( {{3 \over 4}} \right) ( \] \Rightarrow