Telescoping: all problems

What is the value of $4 \cdot (-1 + 2 - 3 + 4 - 5 + 6 - 7 + \cdots + 1000)$?

What is the value of the expression $\frac{1 \cdot 2 \cdot 3 \cdot 4 \cdot 5 \cdot 6 \cdot 7 \cdot 8}{1+2+3+4+5+6+7+8}$?

What is the value of the product $\left(1+\frac{1}{1}\right) \cdot \left(1+\frac{1}{2}\right) \cdot \left(1+\frac{1}{3}\right) \cdot \left(1+\frac{1}{4}\right) \cdot \left(1+\frac{1}{5}\right) \cdot \left(1+\frac{1}{6}\right)$?

Find the value of the expression $100 - 98 + 96 - 94 + 92 - 90 + \cdots + 8 - 6 + 4 - 2$.

What is the value of the product $\left(\frac{1 \cdot 3}{2 \cdot 2}\right)\left(\frac{2 \cdot 4}{3 \cdot 3}\right)\left(\frac{3 \cdot 5}{4 \cdot 4}\right) \cdots$ $\left(\frac{97 \cdot 99}{98 \cdot 98}\right)\left(\frac{98 \cdot 100}{99 \cdot 99}\right)$?

What is the value of $1 + 3 + 5 + \cdots + 2017 + 2019 - 2 - 4 - 6 - \cdots - 2016 - 2018$?

What is the product of $\frac{3}{2} \times \frac{4}{3} \times \frac{5}{4} \times \cdots \times \frac{2006}{2005}$?

What is the value of $\frac{1}{3} \cdot \frac{2}{4} \cdot \frac{3}{5} \cdots \frac{18}{20} \cdot \frac{19}{21} \cdot \frac{20}{22}$?

If $n = 10^{2020} - 10^{2019} + 10^{2018} - 10^{2017} + \cdots + 10^2 - 10^1$ what is the sum of digits of the integer $n$?

Evaluate $\frac{1}{2 \times 3} + \frac{1}{3 \times 4} + \cdots + \frac{1}{10 \times 11}$

Evaluate $1 - 2 - 3 + 4 + 5 + 6 - 7 - 8 - 9 - 10 \cdots + 90 + 91$ (1 positive consecutive value, 2 negative consecutive values, 3 positive consecutive values, …)

Let the sequence $a_n$ be defined as $a_n = 3^{1+2+3+\cdots+n}$ for all positive integers $n$. Sohil is trying to evaluate the following sequence: $\frac{1}{2} \times \frac{2}{3} \times \frac{3}{4} \times \ldots$. Due to his bad handwriting, in the expression above, he accidentally writes each multiplication sign before all fractions of the form $\frac{a_n}{a_n+1}$ (for example, $\frac{3^1}{3^1+1}$, $\frac{3^{(1+2)}}{3^{(1+2)}+1}$, etc.) with a plus sign. What is the value of the new expression?