Evaluate the Summation sum from n=4 to 7 of 2n

Solution:

Step 1:

Write out the terms of the series for each integer $n$ from $4$ to $7$.

$2\cdot 4+2\cdot 5+2\cdot 6+2\cdot 7$

Step 2:

Proceed to simplify the expression.

Step 2.1:

Calculate $2$ times $4$.

$8+2\cdot 5+2\cdot 6+2\cdot 7$

Step 2.2:

Calculate $2$ times $5$.

$8+10+2\cdot 6+2\cdot 7$

Step 2.3:

Combine $8$ and $10$.

$18+2\cdot 6+2\cdot 7$

Step 2.4:

Calculate $2$ times $6$.

$18+12+2\cdot 7$

Step 2.5:

Combine $18$ and $12$.

$30+2\cdot 7$

Step 2.6:

Calculate $2$ times $7$.

$30+14$

Step 2.7:

Combine $30$ and $14$.

$44$

The final result is $44$.

Knowledge Notes:

To evaluate the summation $\sum _{n=4}^{7}2n$, we are essentially adding up the values of the function $2n$ for each integer $n$ from $4$ to $7$. The process involves the following steps:

1. Expansion: We start by expanding the summation into its individual terms by substituting each value of $n$ into the function $2n$.

2. Simplification: This involves performing the arithmetic operations in the expanded form, step by step. We multiply each term by $2$ and then add the results together.

3. Arithmetic Operations: We use basic arithmetic operations, such as multiplication and addition, to simplify the expression. Multiplication is done before addition according to the order of operations (PEMDAS/BODMAS).

4. Final Summation: After simplifying each term, we add them together to find the final sum.

In this particular problem, the function is linear, and the summation is a finite arithmetic series. The general formula for the sum of an arithmetic series is not required here, as we can directly compute the sum by expanding and simplifying the terms. However, for larger series or more complex patterns, knowing the formula for the sum of an arithmetic series can be very helpful:

$$S_n=\frac{n}{2}(a_1+a_n)$$

where $S_n$ is the sum of the first $n$ terms of the series, $a_1$ is the first term, and $a_n$ is the nth term. In this case, the series is simple enough to calculate without the formula.