Evaluate the Summation sum from i=1 to 10 of 9i-3

Solution:

Step 1

Decompose the given summation into two separate summations using the distributive property of summation: $\sum_{i = 1}^{10} (9i - 3) = 9\sum_{i = 1}^{10} i - \sum_{i = 1}^{10} 3$

Step 2

Calculate $9\sum_{i = 1}^{10} i$.

Step 2.1

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

Step 2.2

Plug in the upper limit of the summation and include the coefficient: $9 \left( \frac{10(10 + 1)}{2} \right)$

Step 2.3

Perform the arithmetic operations.

Step 2.3.1

Combine $10$ and $1$: $9 \times \frac{10 \cdot 11}{2}$

Step 2.3.2

Calculate $10 \times 11$: $9 \left( \frac{110}{2} \right)$

Step 2.3.3

Divide $110$ by $2$: $9 \times 55$

Step 2.3.4

Multiply $9$ by $55$: $495$

Step 3

Compute $\sum_{i = 1}^{10} -3$.

Step 3.1

Apply the constant series sum formula: $\sum_{i = 1}^{n} c = c \cdot n$

Step 3.2

Insert the given values: $(-3) \cdot 10$

Step 3.3

Calculate the product of $-3$ and $10$: $-30$

Step 4

Combine the results from the two summations: $495 - 30$

Step 5

Subtract $30$ from $495$ to get the final result: $465$

Knowledge Notes:

The problem involves evaluating a finite arithmetic series, which is a summation of terms in a sequence where each term after the first is obtained by adding a constant difference to the previous term. The specific series in question is a linear combination of an arithmetic series and a constant series.

Key points to remember when solving such problems include:

1. Distributive Property of Summation: This property allows us to split a summation of terms into separate summations if the terms are being added or subtracted. For example, $\sum (a_i + b_i) = \sum a_i + \sum b_i$.

2. Arithmetic Series Sum Formula: 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 from the observation that the sum of the first and the last term, the second and the second-to-last term, and so on, are all equal to $n + 1$, and there are $\frac{n}{2}$ such pairs.

3. Constant Series Sum Formula: The sum of a constant $c$ over $n$ terms is simply $c$ multiplied by the number of terms: $\sum_{i = 1}^{n} c = c \cdot n$.

4. Simplification: After applying the formulas, simplifying the expressions step by step is crucial to avoid mistakes and to make the calculation easier.

5. Combining Results: After evaluating each part of the summation separately, the final step is to combine the results to get the answer to the original problem.