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Find the value of 1/(81^n)-((10)/(81^n))...

Find the value of `1/(81^n)-((10)/(81^n))^(2n)C_1+((10^2)/(81^n))^(2n)C_2-((10^3)/(81^n))^(2n)C_3++(10^(2n))/(81^n)` .

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Verified by Experts

The correct Answer is:
`1`
We have
`(1)/(81^(n)) - (10)/(81^(n)).^(2n)C_(1)+(10^(2))/(81^(n)).^(2n)C_(2)- (10^(3))/(81^(n)) .^(2n)C_(3)+"……."+(10^(2n))/(81^(n))`
`= 1/(81^(n))[.^(2n)C_(0) - .^(2n)C_(1)10^(1) + .^(2n)C_(2)10^(2)-.^(2n)C_(3)10^(3)+"……"+.^(2n)C_(2n)10^(2n)]`
`= (1)/(81^(n))[1-10]^(2n)`
`= ((-9)^(2n))/(81^(n))= (81^(n))/(81^(n)) = 1`
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