List of top Mathematics Questions on Sequences and Series

Consider sequences of positive real numbers of the form x,2000,y,… x, 2000, y, \dots , in which every term after the first is 1 less than the product of its two immediate neighbors. For how many different values of x x does the term 2001 appear somewhere in the sequence?
If the numbers a1,a2,…,an a_1, a_2, \ldots, a_n are different from zero and form an arithmetic progression, then 1a1+1a2+1a3+⋯+1an \frac{1}{a_1} + \frac{1}{a_2} + \frac{1}{a_3} + \dots + \frac{1}{a_n} is equal to 
If ∣a∣<1 |a| < 1 and ∣b∣<1 |b| < 1 , then the sum of the series a(a+b)+a2(a2+b2)+a3(a3+b3)+⋯ a(a + b) + a^2(a^2 + b^2) + a^3(a^3 + b^3) + \cdots  
lim⁡n→∞[5−15+125−⋯+(−1)n−1⋅15n] \lim_{n \to \infty} \left[ 5 - \frac{1}{5} + \frac{1}{25} - \dots + (-1)^{n-1} \cdot \frac{1}{5^n} \right]