List of top Mathematics Questions on Sequence and Series

$\sum_{r=1}^{n} (r \cdot r!) = $ ________

If we insert two numbers between $\sqrt{2}$ and $4$ so that the resulting sequence is in G.P., then the inserted numbers in the order are

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:}

There are rooms numbered from 1 to 99 in an apartment. A mathematician notices that the sum of all room numbers before his room is equal to the sum of all room numbers after his room. Find his room number.

There are 49 houses in a row, numbered consecutively. A scientist finds a magical house number such that the sum of house numbers on its left is equal to the sum on its right. Find the magical house number.

A company adds storage: (2^4) TB (1st month), (2^5) (2nd), (2^6) (3rd)... Total additional storage from 4th to n-th month is:

Houses are numbered from 1 to 49. Find the house number such that the sum of numbers before it equals the sum after it.

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 $