Let n be a positive integer. Let A = Σ k = 0 ⁿ (- 1)^(k) ⁿ C k [ (1/2)^(k) + (3/4)^(k) + (7/8)^(k) + (15/16)^(k) + (31/32)^(k) ]. If 63A = 1 - 1/(2³⁰), then n is equal to _ _ _ _ _ _ _.

Binomial Theorem and Its Simple Applications: JEE Main 2021 Mathematics question

JEE Main 2021 · 16 Mar, Evening shift · Q31

JEE Main 202116 Mar, Evening shiftMathematicsHardBinomial Theorem and Its Simple Applications131× since 2002

Let n be a positive integer. Let

A = \sum\limits_{k = 0}^n {{{( - 1)}^k}{}^n{C_k}\left[ {{{\left( {{1 \over 2}} \right)}^k} + {{\left( {{3 \over 4}} \right)}^k} + {{\left( {{7 \over 8}} \right)}^k} + {{\left( {{{15} \over {16}}} \right)}^k} + {{\left( {{{31} \over {32}}} \right)}^k}} \right]}

. If

63A = 1 - {1 \over {{2^{30}}}}

, then n is equal to _____________.

Your answerType your answer. Integers and decimals are accepted.

JEE Main 2021 · 16 Mar, Evening shift · Q31

JEE Main 202116 Mar, Evening shiftMathematicsHardBinomial Theorem and Its Simple Applications131× since 2002

Let n be a positive integer. Let

A = \sum\limits_{k = 0}^n {{{( - 1)}^k}{}^n{C_k}\left[ {{{\left( {{1 \over 2}} \right)}^k} + {{\left( {{3 \over 4}} \right)}^k} + {{\left( {{7 \over 8}} \right)}^k} + {{\left( {{{15} \over {16}}} \right)}^k} + {{\left( {{{31} \over {32}}} \right)}^k}} \right]}

. If

63A = 1 - {1 \over {{2^{30}}}}

, then n is equal to _____________.

Your answerType your answer. Integers and decimals are accepted.