Evaluate the Summation sum from i=1 to 6 of 7i-2

Solution:

Step 1:

Decompose the given summation into two separate summations that are easier to handle. $\sum_{i = 1}^{6} (7i - 2) = 7\sum_{i = 1}^{6} i - \sum_{i = 1}^{6} 2$

Step 2:

Determine the value of $7 \sum_{i = 1}^{6} i$.

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: $7 \left( \frac{6(6 + 1)}{2} \right)$

Step 2.3:

Perform the arithmetic operations.

Step 2.3.1:

Add together $6$ and $1$: $7 \left( \frac{6 \cdot 7}{2} \right)$

Step 2.3.2:

Calculate $6 \times 7$: $7 \left( \frac{42}{2} \right)$

Step 2.3.3:

Divide $42$ by $2$: $7 \cdot 21$

Step 2.3.4:

Multiply $7$ by $21$: $147$

Step 3:

Compute the summation of the constant term $\sum_{i = 1}^{6} -2$.

Step 3.1:

Apply the formula for the sum of a constant series: $\sum_{i = 1}^{n} c = cn$

Step 3.2:

Insert the values into the formula: $(-2)(6)$

Step 3.3:

Multiply $-2$ by $6$: $-12$

Step 4:

Combine the results from the two summations: $147 + (-12)$

Step 5:

Subtract $12$ from $147$ to get the final result: $135$

Knowledge Notes:

The problem at hand involves evaluating a summation expression, specifically the sum of a linear function of $i$, from $i=1$ to $i=6$. The problem-solving process breaks down the summation into more manageable parts and applies known formulas for arithmetic series and constant series.

1. Summation of a Linear Function: The summation of a linear function can be split into the sum of its individual terms. In this case, the linear function $7i - 2$ is split into $7\sum i$ and $\sum -2$.

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 is equal to the average of the first and last term, multiplied 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, so the sum is the constant multiplied by the number of terms.

4. Combining Summations: After evaluating each summation separately, the results are combined to give the final sum. This involves basic arithmetic operations such as addition and subtraction.

5. Order of Operations: When simplifying expressions, it is important to follow the correct order of operations, which typically is parentheses, exponents, multiplication and division (from left to right), and addition and subtraction (from left to right).

In this problem, the process involves decomposing the original summation, applying the appropriate formulas for each part, performing the necessary arithmetic operations, and finally combining the results to obtain the final answer.