IIMB 2002 (QA) PYQs

The number of terms common to the two A.P.'s $3,7,11,........,407$ and $2,9,16,..........., 709$ is

The sum of the first 3 terms of an AP is 6 and that of the last 3 terms is 16. If the AP has 13 terms, what is the Arithmetic mean of the three terms that are exactly at the middle of the progression?

What is the sum of the infinite series: $1 - \frac{1}{2} - \frac{1}{4} + \frac{1}{8} - \frac{1}{16} - \frac{1}{32} + \frac{1}{64} - \frac{1}{128} - \frac{1}{256} + \dots?$