Evaluate the Summation sum from n=1 to 80 of 7-1/2n
The given problem is to perform a mathematical calculation. Specifically, it involves a summation (also known as a series), where you are asked to evaluate the total sum starting from n=1 and ending at n=80 of the expression 7 - 1/(2n). The term 7 is a constant that will be included in each summand, while the term 1/(2n) represents a sequence with a variable denominator that changes with each increment of the variable n. The task involves calculating the sum of all the individual terms produced by plugging in the values of n from 1 to 80 into the given expression and adding them all together to obtain a final result.
$\sum_{n = 1}^{80} 7 - \frac{1}{2} n$
Answer
Solution:
Step 1: Simplify the given summation expression.
Step 1.1: Combine the constant $7$ with the term $-\frac{1}{2}n$ to form the expression $7 - \frac{n}{2}$.
Step 1.2: Express the summation as $\sum_{n = 1}^{80} (7 - \frac{n}{2})$.
Step 2: Decompose the summation into two separate summations.
- The original summation is broken down as follows: $\sum_{n = 1}^{80} (7 - \frac{n}{2}) = \sum_{n = 1}^{80} 7 + \sum_{n = 1}^{80} (-\frac{n}{2})$.
Step 3: Calculate the summation of the constant term $7$.
Step 3.1: Use the summation formula for a constant: $\sum_{k = 1}^{n} c = cn$.
Step 3.2: Plug in the values: $(7)(80)$.
Step 3.3: Perform the multiplication: $7 \times 80 = 560$.
Step 4: Evaluate the summation involving the variable $n$.
Step 4.1: Factor out $-\frac{1}{2}$ from the summation: $-\frac{1}{2} \sum_{n = 1}^{80} n$.
Step 4.2: Apply the formula for the summation of the first $n$ natural numbers: $\sum_{k = 1}^{n} k = \frac{n(n + 1)}{2}$.
Step 4.3: Substitute the values and include the factor from Step 4.1: $-\frac{1}{2} \left( \frac{80(80 + 1)}{2} \right)$.
Step 4.4: Simplify the expression:
Step 4.4.1: Combine $80$ and $1$: $-\frac{1}{2} \cdot \frac{80 \cdot 81}{2}$.
Step 4.4.2: Cancel out the common factor of $2$:
Step 4.4.2.1: Rewrite the negative factor: $\frac{-1}{2} \cdot \frac{6480}{2}$.
Step 4.4.2.2: Factor out $2$ from $6480$: $\frac{-1}{2} \cdot \frac{2(3240)}{2}$.
Step 4.4.2.3: Cancel the $2$s: $\frac{-1}{\cancel{2}} \cdot \frac{\cancel{2}(3240)}{2}$.
Step 4.4.2.4: Finalize the expression: $-1 \left( \frac{3240}{2} \right)$.
Step 4.4.3: Complete the simplification:
Step 4.4.3.1: Divide $3240$ by $2$: $-1 \cdot 1620$.
Step 4.4.3.2: Multiply by $-1$: $-1620$.
Step 5: Combine the results of the two summations.
- Add the outcomes from Step 3.3 and Step 4.4.3.2: $560 - 1620$.
Step 6: Final calculation.
- Subtract $1620$ from $560$ to get the final result: $-1060$.
Knowledge Notes:
Summation of a constant: The sum of a constant $c$ added up $n$ times is given by the formula $\sum_{k = 1}^{n} c = cn$.
Summation of the first $n$ natural numbers: The sum of the first $n$ natural numbers is calculated using the formula $\sum_{k = 1}^{n} k = \frac{n(n + 1)}{2}$.
Distributive property: When a summation involves a sum or difference, it can be split into separate summations.
Factoring out constants: Constants can be factored out of a summation, which simplifies the calculation.
Cancellation: When a common factor appears in both the numerator and denominator, it can be canceled out to simplify the expression.
Arithmetic operations: Basic arithmetic operations, such as addition, subtraction, multiplication, and division, are used to simplify and calculate the final result of the summation.
