Evaluate the Summation sum from i=1 to 1000 of 2i

Solution:

Step 1:

Extract the constant $2$ from the summation: $2\sum _{i=1}^{1000}i$

Step 2:

Apply the arithmetic series sum formula: $\sum _{i=1}^{n}i=\frac{n(n+1)}{2}$

Step 3:

Insert $n=1000$ into the formula and remember to include the constant from Step 1: $2\times \frac{1000(1000+1)}{2}$

Step 4:

Proceed with the simplification:

Step 4.1:

First, simplify the terms inside the brackets:

$2\times \frac{1000\times 1001}{2}$

Step 4.1.1:

Multiply $1000$ by $1001$:

$2\times \frac{1001000}{2}$

Step 4.2:

Next, reduce the fraction by eliminating the common factor of $2$:

Step 4.2.1:

Eliminate the $2$ from the numerator and denominator:

$\cancel{2}\times \frac{1001000}{\cancel{2}}$

Step 4.2.2:

Conclude with the final expression:

$1001000$

Knowledge Notes:

The problem involves evaluating the summation of a sequence of numbers, specifically the first 1000 positive integers, each multiplied by 2. The solution leverages the formula for the sum of an arithmetic series, which is a sequence of numbers with a constant difference between consecutive terms. The formula for the sum of the first $n$ natural numbers is given by:

$$\sum _{i=1}^{n}i=\frac{n(n+1)}{2}$$

This formula is derived by pairing terms from the beginning and end of the sequence, which always sum to $n+1$, and noticing that there are $\frac{n}{2}$ such pairs.

In the solution, we first factor out the constant multiplier (Step 1), which is a common technique in summation problems to simplify the expression. We then apply the arithmetic series sum formula (Step 2), substitute the specific value for $n$ (Step 3), and simplify the resulting expression (Step 4).

The simplification process involves basic arithmetic operations: addition, multiplication, and division, including canceling out common factors to reduce the expression to its simplest form. The final result gives us the sum of the series.