Let i = √(- 1). If ((- 1 + i 3)²¹)/((1 - i)²⁴) + ((1 + i 3)²¹)/((1 + i)²⁴) = k, and n = [|k|] be the greatest integral part of | k |. Then Σ j = 0 ⁿ⁺⁵ (j + 5)² - Σ j = 0 ⁿ⁺⁵ (j + 5) is equal to _ _ _ _ _.

Complex Numbers and Quadratic Equations: JEE Main 2021 Mathematics question

JEE Main 2021 · 24 Feb, Evening shift · Q33

JEE Main 202124 Feb, Evening shiftMathematicsMediumComplex Numbers and Quadratic Equations264× since 2002

Let

i = \sqrt { - 1}

. If

{{{{\left( { - 1 + i\sqrt 3 } \right)}^{21}}} \over {{{(1 - i)}^{24}}}} + {{{{\left( {1 + i\sqrt 3 } \right)}^{21}}} \over {{{(1 + i)}^{24}}}} = k

, and

n = [|k|]

be the greatest integral part of | k |. Then

\sum\limits_{j = 0}^{n + 5} {{{(j + 5)}^2} - \sum\limits_{j = 0}^{n + 5} {(j + 5)} }

is equal to _________.

Your answerType your answer. Integers and decimals are accepted.