+54 1.)
The first term of a sequence is 2, and each subsequent term is the reciprocal of the square of the preceding term. What is the positive square root of the fifth term?
2.)
Calculate
\[
\left(1 - \frac12 \right) \cdot \left(1 - \frac13 \right) \cdot \left(1 - \frac14\right) \dotsm \left(1 - \frac {1}{2009} \right).
\]
3.)
What is the maximum number of consecutive positive integers that can be added together to create a sum less than 400?
4.)
What is the units digit of $2^{1993} + 3^{1993}$?
5.)
What is the value of the sum
$\dfrac {1}{1\cdot 3} + \dfrac {1}{3\cdot 5} + \dfrac {1}{5\cdot 7} + \dfrac {1}{7\cdot 9} + \cdots + \dfrac {1}{199\cdot 201}$
? Express your answer as a fraction in simplest form.
6.)
Compute
\[1 \cdot \frac {1}{2} + 2 \cdot \frac {1}{4} + 3 \cdot \frac {1}{8} + \dots + n \cdot \frac {1}{2^n} + \dotsb.\]
