Evaluate the Summation sum from i=1 to 10 of -3i+7

Solution:

Step 1:

Decompose the given summation into two separate summations according to the distributive property of summation.

$$\sum_{i = 1}^{10} (-3i + 7) = -3 \sum_{i = 1}^{10} i + \sum_{i = 1}^{10} 7$$

Step 2:

Calculate the summation of the first term $-3 \sum_{i = 1}^{10} i$.

Step 2.1:

Use the arithmetic series sum formula:

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

Step 2.2:

Insert the upper limit of the summation into the formula and include the coefficient:

$$-3 \left( \frac{10(10 + 1)}{2} \right)$$

Step 2.3:

Proceed with the simplification:

Step 2.3.1:

Combine the numbers inside the parentheses:

$$-3 \frac{10 \cdot 11}{2}$$

Step 2.3.2:

Perform the multiplication inside the parentheses:

$$-3 \left( \frac{110}{2} \right)$$

Step 2.3.3:

Divide $110$ by $2$:

$$-3 \cdot 55$$

Step 2.3.4:

Finally, multiply $-3$ by $55$:

$$-165$$

Step 3:

Compute the summation of the constant term $\sum_{i = 1}^{10} 7$.

Step 3.1:

Apply the sum formula for a constant sequence:

$$\sum_{i = 1}^{n} c = cn$$

Step 3.2:

Substitute the constant value and the upper limit of the summation:

$$(7)(10)$$

Step 3.3:

Carry out the multiplication:

$$70$$

Step 4:

Combine the results from the two summations:

$$-165 + 70$$

Step 5:

Add the two results together to get the final answer:

$$-95$$

Knowledge Notes:

To solve the given problem, we used several key concepts in summation and arithmetic series:

1. Distributive Property of Summation: This property allows us to split a summation of terms into separate summations, which can be easier to evaluate. For example, $\sum (a + b) = \sum a + \sum b$.

2. 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 formula is derived from the fact that the sum of an arithmetic series can be found by multiplying the average of the first and last terms by the number of terms.

3. Summation of a Constant: The sum of a constant $c$ over $n$ terms is simply $cn$. This is because each term in the summation is the same, and there are $n$ such terms.

4. Simplification: After applying the formulas, we simplified the expressions step by step, performing basic arithmetic operations such as addition, multiplication, and division.

By understanding and applying these concepts, we were able to evaluate the given summation efficiently.