_{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

\[1 \cdot \frac {1}{2} + 2 \cdot \frac {1}{4} + 3 \cdot \frac {1}{8} + \dots + n \cdot \frac {1}{2^n} + \dotsb.\]

Jdaye Jan 27, 2018