Evaluate the Summation sum from n=13 to 16 of n-4
The question asks you to calculate the total sum of a sequence of numbers derived from a given expression, which in this case is "n-4". You are supposed to evaluate this expression for each integer value of 'n' starting from 13 and ending at 16. For each of these values of 'n', you subtract 4 as indicated by the expression and then add all these results together to find the final summation.
$\sum_{n = 13}^{16} n - 4$
Answer
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:
Expansion: We expand the summation by calculating each term separately. This is done by substituting the values of $n$ into the formula $n - 4$.
Simplification: We simplify the expanded series by performing the arithmetic operations step by step. This involves basic subtraction and addition.
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.
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.
