I term these as High-Level Illustrations:  Just going through the solutions of these questions > will Create a sense of sense of wonder in you. I have spent 6 months of my life in this wanderlust | My approach of interacting with a question is never solving it down rather living it, Tearing it apart in as many as possible ways I can

Here I have delivered my sense of understanding to Indian Students who like to do complex math in a simplified way

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1) Decoding pattern by observing the calculated cases manually one by one

2) Using the factorized formulae of

A^3+B^3+C^3 - 3ABC to prove observed and initially assumed pattern

This question was asked in PUTNAM 2019,  There are Few questions in putnam which I find extremely relevant for JEE Aspirants which I have explained in my videos.

> This question gives you an insight of EXTRACTING PATTERNS OUT OF RANDOMNESS.

> Generally we Try to find what lies in the domain of the expression but in this question The approach is extracting What's not In it's domain

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1) Decoding pattern ( Observation Based )

2) Basic Trigonometric

3) Properties of cosines playing EVEN - ODD ​

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Again, This Question was asked in PUTNAM 1985,  There are Few questions in putnam which I find extremely relevant for JEE Aspirants which I have explained in my videos.

> Like the first one, This Question Gives You  an Insight of Extracting Patterns out of randomness -> This one thing you will find in many of my illustrations because How the one who solved these amazes me to have my hands on such questions

> In trigonometry we try to solve a question by deducing the expression but this general approach will not work here. One needs to have very keen observation and familiarity with the approach of hidden patterns

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Take Away

1) '3' - The number 3 could lead us to the concept if one has practiced enough of complex numbers.

2) Factorials in denominator was another clue that there may be an interference of the number 'e' which again leads us to the use of complex numbers

I found this question back in those days when I was Preparing For JEE On Brilliant.

Questions which Include factorial Manipulation are generally outstanding > You will find many in Sequence & Series Chapter but this one got my attention because it was supposed to be solved by Using Complex numbers

After going through it I tried to decode that what path Must  be followed in order to crack it, These 3 elements 3, !, e > If we carefully look at this question  it looks the interference of these 3 terms which will lead us to our answer