Evaluate the Summation sum from i=1 to 10 of 5i

Solution:

Step 1:

Extract the constant $5$ from the summation: $5 \sum_{i = 1}^{10} i$

Step 2:

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

Step 3:

Insert $n=10$ into the formula and don't forget to include the constant we factored out: $5 \left( \frac{10(10 + 1)}{2} \right)$

Step 4:

Proceed with the simplification:

Step 4.1:

Combine $10$ and $1$: $5 \times \frac{10 \times 11}{2}$

Step 4.2:

Calculate $10 \times 11$: $5 \left( \frac{110}{2} \right)$

Step 4.3:

Divide $110$ by $2$: $5 \times 55$

Step 4.4:

Finally, multiply $5$ by $55$: $275$

Knowledge Notes:

The problem involves evaluating the summation of a linear sequence where each term is a multiple of the index $i$. The process of solving this problem includes several key knowledge points:

1. Factorization: This is the process of taking out a common factor from all terms of a summation or expression. In this case, the number $5$ is factored out.

2. Summation Formula for Arithmetic Series: The summation 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 for the sum of an arithmetic series where the difference between consecutive terms is constant (in this case, the difference is $1$).

3. Substitution: This involves replacing a variable in an algebraic expression with its value. Here, we substitute $n$ with $10$ in the summation formula.

4. Arithmetic Operations: The final steps involve basic arithmetic operations such as addition, multiplication, and division to simplify the expression and find the sum.

By understanding these concepts, one can solve similar problems involving summation of arithmetic sequences.