Evaluate the Summation sum from i=1 to 60 of 2i
The problem asks for the evaluation of a mathematical summation where the summation index $i$runs from 1 to 60, and for each value of $i$, the term to be added is $2i$. The task is to calculate the total sum of all these terms. The term $2i$indicates that each term in the summation is twice the value of the current index $i$. To solve the problem, one would typically use the formula for the sum of an arithmetic series or manually add the terms if a calculator with summation capabilities is available.
$\sum_{i = 1}^{60} 2 i$
Answer
Solution:
Step 1:
Extract the constant $2$ from the summation: $2 \sum_{i = 1}^{60} i$
Step 2:
Apply the arithmetic series sum formula: $\sum_{i = 1}^{n} i = \frac{n(n + 1)}{2}$
Step 3:
Insert the upper limit of the summation into the formula, and don't forget to include the extracted constant: $2 \left( \frac{60(60 + 1)}{2} \right)$
Step 4:
Proceed with the simplification process.
Step 4.1:
Begin simplifying the terms within the expression.
Step 4.1.1:
Combine $60$ and $1$: $2 \cdot \frac{60 \cdot 61}{2}$
Step 4.1.2:
Perform the multiplication of $60$ and $61$: $2 \left( \frac{3660}{2} \right)$
Step 4.2:
Eliminate the common factor of $2$.
Step 4.2.1:
Remove the common factor: $\cancel{2} \left( \frac{3660}{\cancel{2}} \right)$
Step 4.2.2:
Finalize the expression: $3660$
Knowledge Notes:
The problem involves evaluating the summation of a sequence of numbers where each term is a multiple of $2$. The process utilizes the properties of summation and arithmetic sequences to simplify the calculation.
Factorization: The constant factor within the summation can be factored out, simplifying the expression and reducing the problem to finding the sum of a simple arithmetic series.
Arithmetic Series Sum Formula: The sum of the first $n$ natural numbers is given by the formula $\sum_{i = 1}^{n} i = \frac{n(n + 1)}{2}$. This is a well-known result in arithmetic series which states that the sum of a sequence of $n$ consecutive natural numbers starting from $1$ is equal to half the product of the last number in the sequence and its successor.
Substitution: After applying the formula, the specific values from the problem are substituted into the formula to calculate the sum.
Simplification: The process involves basic arithmetic operations such as addition, multiplication, and division, as well as simplifying by canceling out common factors.
Final Answer: The result of the summation is obtained after the simplification steps, providing the final answer to the problem.
