Evaluate (3+1)(3²+1)(3⁴+1)(3⁸+1)(3¹⁶+1)(3³²+1).

Evaluate \(\left(1-\dfrac{1}{2^2}\right)\left(1-\dfrac{1}{3^2}\right)\left(1-\dfrac{1}{4^2}\right)...\left(1-\dfrac{1}{1999^2}\right)\left(1-\dfrac{1}{2000^2}\right)\)

[Hint: a² - b² = (a-b)(a+b)]

Find the last 4 digits in the addition of

1 + 11 + 111 + 1111 + 11111 + 111111 + ...+ 111...111(1004 1s).

Evaluate \(\dfrac{1}{1+2}+\dfrac{1}{1+2+3}+\dfrac{1}{1+2+3+4}+..+\dfrac{1}{1+2+3+...+50}\)

Evaluate 

\(\left[1\times\left(1-\dfrac{1}{2}\right)\times\left(1-\dfrac{1}{3}\right)\times\left(1-\dfrac{1}{4}\right)\times\left(1-\dfrac{1}{5}\right)\times...\times\left(1-\dfrac{1}{2000}\right)\right]\times2000\)

Evaluate \(\left(1+\dfrac{1}{53}+\dfrac{1}{59}+\dfrac{1}{61}\right)\times\left(\dfrac{1}{53}+\dfrac{1}{59}+\dfrac{1}{61}+\dfrac{1}{67}\right)\times\left(1+\dfrac{1}{53}+\dfrac{1}{59}+\dfrac{1}{61}+\dfrac{1}{67}\right)\times\left(\dfrac{1}{53}+\dfrac{1}{59}+\dfrac{1}{61}\right)\)

Evaluate \(\dfrac{1}{2}+\left(\dfrac{1}{3}+\dfrac{2}{3}\right)+\left(\dfrac{1}{4}+\dfrac{2}{4}+\dfrac{3}{4}\right)+\left(\dfrac{1}{5}+\dfrac{2}{5}+\dfrac{3}{5}+\dfrac{4}{5}\right)+...+\left(\dfrac{1}{100}+\dfrac{2}{100}+\dfrac{3}{100}+...+\dfrac{99}{100}\right)\)

What is the of n in \(\dfrac{1}{2}+\dfrac{1}{6}+\dfrac{1}{12}+...+\dfrac{1}{n\left(n+1\right)}=\dfrac{1999}{2000}?\)

Evaluate 1+2+2²+2³+2⁴+...+2²⁰⁰⁰.

Evaluate 2000 x \(\left(1+\dfrac{1}{2}\right)\)x\(\left(1+\dfrac{1}{3}\right)\)x\(\left(1+\dfrac{1}{4}\right)\)x...x\(\left(1+\dfrac{1}{2000}\right)\)

The value of (100 + 99 + 98 - 97 - 96 + 95 + 94 + 93 - 92 - 91 + .. + 10 + 9 + 8 - 7 - 6 + 5 +4 + 3 - 2 - 1 ) is

Evaluate \(\dfrac{1}{1\times4}+\dfrac{1}{4\times7}+\dfrac{1}{7\times10}+...+\dfrac{1}{67\times70}\)

It given \(A=\dfrac{3000\times3003}{3001\times3002}\),\(B=\dfrac{3000\times3002}{3001\times3003}\)and \(C=\dfrac{3000\times3001}{3002\times3003}\), then

(A) C < B < A

(B) A < C < B

(C) C < A < B

(D) A < B < C

(E) B < A < C

If m + |m| + n = 8 and  |n| + m-n = 9, find m-n.

Find the smallest value of |a-1000| + |a-1001|.

Simplify |a+1| + |a-1| for -1 <= a <=0.