Evaluate the Summation sum from i=1 to 1000 of 2i
The problem is asking for the calculation of a particular summation, which is a mathematical expression representing the addition of a sequence of terms. Specifically, this summation is defined to start with the variable 'i' at 1 and continue by incrementing 'i' by 1 each step until 'i' reaches 1000. At each step, the current value of 'i' is multiplied by 2 and added to the running total. The question is requiring the evaluation of this sum to find a single numerical value that represents the total of all these 2i terms for 'i' ranging from 1 to 1000.
$\sum_{i = 1}^{1000} 2 i$
Answer
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.
