UGAT Bangalore Algebra > Progression & Series PYQs

Let $A = \{a_1, a_2, ..., a_i, ...\}$ be an arithmetic progression, and let $B = \{b_1, b_2, ..., b_i, ...\}$ be a geometric progression. The common difference for A is 2. The common ratio for B is 0.2, and $b_1 = 0.8$ The infinite sum of the products $a_i b_i$ is 1, where $i = 1, 2, 3, ...$ What is $a_i$?

In a geometric progression, the first term is $a$ (with $a \ne 0$), the common ratio is $r$ (with $r < 1$), and the sum to infinity is $S$. If the second term of this geometric progression is equal to the negative of the sum to infinity, what is the value of $[r]$, where $[r]$ denotes the greatest integer function?