Evaluate the Summation sum from n=13 to 16 of n-4

Solution:

Step 1:

Write out each term of the summation by plugging in the values of $n$ from $13$ to $16$.

$$13-4+14-4+15-4+16-4$$

Step 2:

Proceed to simplify the expression.

Step 2.1:

Deduct $4$ from $13$ to get $9$.

$$9+14-4+15-4+16-4$$

Step 2.2:

Next, subtract $4$ from $14$ to obtain $10$.

$$9+10+15-4+16-4$$

Step 2.3:

Combine $9$ and $10$ to sum up to $19$.

$$19+15-4+16-4$$

Step 2.4:

Take away $4$ from $15$ resulting in $11$.

$$19+11+16-4$$

Step 2.5:

Add together $19$ and $11$ to reach $30$.

$$30+16-4$$

Step 2.6:

Subtract $4$ from $16$ to get $12$.

$$30+12$$

Step 2.7:

Finally, add $30$ and $12$ to find the sum, which is $42$.

$$42$$

Knowledge Notes:

The problem involves evaluating a finite summation, which is a common concept in algebra and calculus. A summation is the operation of adding a sequence of numbers; the result is their sum or total. The notation $\sum$ represents the summation operator. The problem provided is a simple arithmetic series where each term is defined by the formula $n-4$, with $n$ taking on values from $13$ to $16$.

To solve the problem, we follow these steps:

1. Expansion: We expand the summation by calculating each term separately. This is done by substituting the values of $n$ into the formula $n-4$.

2. Simplification: We simplify the expanded series by performing the arithmetic operations step by step. This involves basic subtraction and addition.

3. Combining like terms: We combine terms that are alike to simplify the expression further. This is a standard algebraic technique used to make calculations more manageable.

4. Final Summation: We add up the simplified terms to get the final result.

In this case, the summation is straightforward because it is a finite series with a simple linear term. No advanced calculus techniques are necessary, and the solution involves elementary arithmetic operations.