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Given  By Binomial Theorem It can also be written as  Now, Again By Binomial Theorem, From (1) and (2) we get,
Given the expression, Binomial expansion of this expression is  Now Applying Binomial Theorem again, And  Now, From (1), (2) and (3) we get,
Given, the expression  Fifth term from the beginning  is  And Fifth term from the end is, Now, As given in the question, So, From Here , From here, Hence the value of n is 10.
As we can write 0.99 as 1-0.01, Hence the value of     is 0.951 approximately.
First, lets simplify the expression  using binomial expansion, And Now, Now, Putting  we get,
First let's simplify the expression  using binomial theorem, So, And  Now, Now, Putting  we get
we need to prove,    where k is some natural number. Now let's add and subtract b from a so that we can prove the above result, Hence, is a factor of .
First, lets expand both expressions individually, So, And  Now, Now, for the coefficient of , we multiply and add those terms whose product gives .So, The term which contain are, Hence the coefficient of  is 171.
As we know the Binomial expansion of  is given by  Given in the question, Now, dividing (1) by (2) we get, Now, Dividing (2) by (3) we get,  Now, From (4) and (5), we get,
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