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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

$\dfrac {1}{1\cdot 3} + \dfrac {1}{3\cdot 5} + \dfrac {1}{5\cdot 7} + \dfrac {1}{7\cdot 9} + \cdots + \dfrac {1}{199\cdot 201}$

6.)

Compute

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

Jan 27, 2018
edited by Jdaye  Jan 28, 2018

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1)

2, 1/4, 16, 1/256, 65,536........etc.

Sqrt(65,536) = 256

3)

[a*(a+1)] / 2 =400, solve for a

a = ~ 27, so that:

1+2+3+.........+27 = 378

4)

2^1993 + 3^1993 =2^1 + 3^1 =5 - last digit !!.

Jan 27, 2018