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Q14.    Prove that $\sum_{r = 0}^n3^r \ ^nC_r = 4^n$

As we know from Binomial Theorem, Here putting a = 3, we get,  Hence Proved.

Q13.    Show that $9^{n+1} - 8n - 9$is divisible by 64, whenever n is a positive integer.

If we want to prove that  is divisible by 64, then we have to prove that   As we know, from binomial theorem,  Here putting x = 8 and replacing m by n+1, we get, Now, Using This, Hence is divisible by 64.

Q12.    Find $(x+1)^6 + (x-1)^6$ . Hence or otherwise evaluate $(\sqrt2+1)^6 + (\sqrt2-1)^6$.

Using Binomial Theorem, the expressions  and  can be expressed as , From Here, Now, Using this, we get

Q11.    Find $(a + b)^4 - (a-b)^4$ . Hence, evaluate$(\sqrt{3} + \sqrt2)^4 - (\sqrt3-\sqrt2)^4$ .

Using Binomial Theorem, the expressions  and  can be expressed as From Here, Now, Using this, we get

Q10.    Using Binomial Theorem, indicate which number is larger (1.1)10000 or 1000.

AS we can write 1.1 as 1 + 0.1, Hence,

Using binomial theorem, evaluate the following:

Q9.    $(99)^5$

As we can write 99 in the form 100-1

Using binomial theorem, evaluate the following:

Q8.    $(101)^4$

As we can write 101 in the form 100+1

Using binomial theorem, evaluate the following:

Q7.    $(102)^5$

As we can write 102 in the form 100+2

Using binomial theorem, evaluate the following:

Q6.    $(96)^3$

As 96 can be written as (100-4);

Expand the expression.

Q5.    $\left(x + \frac{1}{x} \right )^6$

Given, The Expression:    the expansion of this Expression is,

Expand the expression.

Q4.    $\left(\frac{x}{3} + \frac{1}{x} \right )^5$

Given, The Expression:    the expansion of this Expression is,

Expand the expression.

Q3.    $(2x-3)^6$

Given, The Expression:    the expansion of this Expression is,

Expand the expression.

Q2.    $\left(\frac{2}{x} - \frac{x}{2} \right )^5$

Given, The Expression:    the expansion of this Expression is,

Expand the expression.

Q1.    $(1-2x)^5$

Given, The Expression:    the expansion of this Expression is,
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