Relation between Coefficient and Indices of x and y

Q. If the coefficients of $5^{th},6^{th}$ and $7^{th}$ terms in the expansion of $(1+x{)}^n$ be in A.P, then n =

1. 7 only
2. 14 only
3. 7 or 14
4. None of these

Q. If ${}^{2n}{C}_6$ is the greatest coefficient in the expansion of ${(1+x)}^{2n},nϵN,$ then $n$bino is

1. 6
2. 7
3. 5
4. 8

Q. The number of positive integers n in the set $\{1,2,3....100\}$ for which the number $\frac{1^2+2^2+3^2+.....n^2}{1+2+3+.....n}$ is an integer is:

1. $33$
2. $34$
3. $50$
4. $100$

Q. If the coefficients of second, third and fourth term in the expansion of $(1+x{)}^{2n}$ are in A.P, then $2n^2-9n+7$ is equal to

1. -1
2. \N
3. 1
4. 3/2

Q. Let $m$ be the smallest positive integer such that the coefficient of $x^2$ in the expansion of $(1+x{)}^2+(1+x{)}^3+...+(1+x{)}^{49}+(1+mx{)}^{50}\text{is}(3n+1{)}^{51}{C}_3$ for some positive interger $n$. Then the value of $n$ is ___

Q. If coefficients of 2, 3 and 4 terms in the binomial expansion of $(1+x{)}^n$ are in A.P, then $n^2-9n$ is equal to

1. -7
2. 7
3. 14
4. -14