Let 3, 6, 9, 12, ....... upto 78 terms and 5, 9, 13, 17, ...... upto 59 terms be two series. Then, the sum of the terms common to both the series is equal to ________.

Let for n = 1, 2, ......, 50, S_n be the sum of the infinite geometric progression whose first term is n² and whose common ratio is $${1 \over {{{(n + 1)}^2}}}$$. Then the value of

$${1 \over {26}} + \sum\limits_{n = 1}^{50} {\left( {{S_n} + {2 \over {n + 1}} - n - 1} \right)} $$ is equal to ___________.

Let A = {1, a₁, a₂ ....... a₁₈, 77} be a set of integers with 1 < a₁ < a₂ < ....... < a₁₈ < 77.

Let the set A + A = {x + y : x, y $$\in$$ A} contain exactly 39 elements. Then, the value of a₁ + a₂ + ...... + a₁₈ is equal to _____________.

If the sum of the first ten terms of the series

$${1 \over 5} + {2 \over {65}} + {3 \over {325}} + {4 \over {1025}} + {5 \over {2501}} + \,\,....$$

is $${m \over n}$$, where m and n are co-prime numbers, then m + n is equal to ______________.