List of top Mathematics Questions on Sequence and Series

Let \( a_n \) denote the term independent of \( x \) in the expansion of \[ \left[x + \frac{\sin(1/n)}{x^2}\right]^{3n}, \] then \[ \lim_{n\to\infty} \frac{(a_n)n!}{\,{}^{3n}P_n} \] equals:
If \( a,b,c \) are in A.P. and the equations \[ (b-c)x^2 + (c-a)x + (a-b) = 0 \] \[ 2(c+a)x^2 + (b+c)x = 0 \] have a common root, then:
If \( (1 + x - 2x^2)^6 = 1 + a_1 x + a_2 x^2 + \cdots + a_{12x^{12} \), then the value of \( a_2 + a_4 + a_6 + \cdots + a_{12} \) is:}
If the sum of \( n \) terms of an A.P. is \( 3n^2 + 5n \) and its \( m \)-th term is 164, then the value of \( m \) is:
The sum of the first four terms of an arithmetic progression is 56. The sum of the last four terms is 112. If its first term is 11, then the number of terms is:
Let \( f(x) \) be a second degree polynomial. If \( f(1)=f(-1) \) and \( p,q,r \) are in A.P., then \( f'(p), f'(q), f'(r) \) are:
Series Expansion $ n^6 + \frac{1}{2} n^4 + \frac{1}{3} n^2 + \cdots + \frac{1}{n} C_n + 1 \quad n \to \infty $