Evaluate the Summation sum from n=17 to 20 of n-4
The given problem is asking for the evaluation of a finite mathematical summation. Specifically, you are to find the total result of adding together the outcomes of the expression "n-4" for each value of 'n' starting from 17 and ending at 20. This means you would calculate the value of this expression when n is 17, then when n is 18, after that when n is 19, and finally when n is 20. These individual results are then to be summed to reach the overall solution to the summation problem.
$\sum_{n = 17}^{20} n - 4$
Answer
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:
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.
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.
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.
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.
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.
