Evaluate the Summation sum from n=17 to 20 of n-4

Solution:

Step 1: Write out the terms of the summation

List each term of the summation by substituting the values of $n$ from $17$ to $20$ into the expression $n-4$.

$17-4+18-4+19-4+20-4$

Step 2: Perform the calculations

Step 2.1: Calculate each term

Deduct $4$ from each value of $n$.

$13+18-4+19-4+20-4$

Step 2.2: Continue with the next term

Repeat the subtraction for the next term.

$13+14+19-4+20-4$

Step 2.3: Combine the first two results

Add the first two results together.

$27+19-4+20-4$

Step 2.4: Proceed with the subtraction

Subtract $4$ from the next term.

$27+15+20-4$

Step 2.5: Sum the current total

Add the current results.

$42+20-4$

Step 2.6: Final subtraction

Subtract $4$ from the last term.

$42+16$

Step 2.7: Sum all terms

Add the remaining terms to get the final result.

$58$

Knowledge Notes:

To evaluate the given summation, we follow a systematic approach:

1. Summation Notation: The summation notation $\sum$ is used to denote the sum of a sequence of terms. The expression $\sum _{n=a}^{b}f(n)$ represents the sum of the function $f(n)$ evaluated at all integers from $a$ to $b$ inclusive.

2. Arithmetic Operations: Basic arithmetic operations are used to simplify the terms within the summation. In this case, we are subtracting a constant number $4$ from each term.

3. Sequential Calculation: The process involves evaluating each term sequentially and simplifying step by step. This helps in avoiding mistakes and keeping track of the calculation.

4. Combining Like Terms: When dealing with summations, it is often useful to combine like terms to simplify the expression. This involves basic addition and subtraction.

5. Final Summation: After simplifying each term, the final step is to add all the simplified terms together to get the sum of the entire series.

In this problem, we are dealing with a finite arithmetic series, which is a sequence of numbers with a constant difference between consecutive terms. Since the difference here is the subtraction of $4$ from each term, the series simplifies to a straightforward addition problem after the constant has been subtracted from each term.