Binomial Theorem · Mathematics · COMEDK

In the expansion $$\left(\frac{1}{x}+x \sin x\right)^{10}, \quad$$ the co - efficient of $$6^{\text {th }}$$ term is equal to $$7 \frac{7}{8}$$, then the principal value of $$x$$ is

The coefficient of the third term in the expansion of $$\left(x^2-\frac{1}{4}\right)^n$$, when expanded in the descending power of $$x$$ is 31, then $$n$$ is

Using mathematical induction, the numbers $$a_n \delta$$ are defined by $$a_0=1, a_{n+1}=3 n^2+n+a_n, (n \geq 0)$$. Then, $$a_n$$ is equal to

If $$49^n+16^n+k$$ is divisible by 64 for $$n \in N$$, then the least negative integral value of $$k$$ is

$$2^{3 n}-7 n-1 \text { is divisible by }$$

The total number of terms in the expansion of $$(x+y)^{60}+(x-y)^{60}$$ is

The coefficient of $$x^{29}$$ in the expansion of $$\left(1-3 x+3 x^2-x^3\right)^{15}$$ is

In the expansion of $$\left(1+3 x+3 x^2+x^3\right)^{2 n}$$, the term which has greatest binomial coefficient, is

If $$U_{n+1}=3 U_n-2 U_{n-1}$$ and $$U_0=2, U_1=3$$, then $$U_n$$ is equal to

If $$4^n+15n+P$$ is divisible by 9 for all $$n\in N$$, then the least negative integral value of P is

The total number of terms in the expansion of $${(x + y)^{100}} + {(x - y)^{100}}$$ is

The coefficient of $${x^{20}}$$ in the expansion of $${(1 + 3x + 3{x^2} + {x^3})^{20}}$$ is

In the expansion of $${(1 - 3x + 3{x^2} - {x^3})^{2n}}$$, the middle term is

Using mathematical induction, the numbers $${a_n}$$'s are defined by $${a_0} = 1,{a_{n + 1}} = 3{n^2} + n + {a_n},(n \ge 0)$$. Then, $${a_n}$$ is equal to

If $$49^n+16n+P$$ is divisible by 64 for all $$n\in N$$, then the least negative integral value of P is

$${2^{3n}} - 7n - 1$$ is divisible by

Number of terms in the binomial expansion of $$(x+a)^{53}+(x-a)^{53}$$ is

The coefficient of $$x^{10}$$ in the expansion of $$1+(1+x)+...+(1+x)^{20}$$ is

Middle term in the expansion of $${\left( {{x^2} + {1 \over {{x^2}}} + 2} \right)^n}$$ is

The ninth term of the expansion $${\left( {3x - {1 \over {2x}}} \right)^8}$$ is