Evaluate the Summation sum from n=5 to 10 of 20-n
The question asks for an evaluation of a specific mathematical expression known as a summation. The summation in question begins with the term at n=5 and concludes with the term at n=10. Each term to be summed is given by the expression '20-n,' which means that for every integer value of n from 5 to 10, you would subtract that value of n from 20, and then add up all of those resulting values to get the final answer.
$\sum_{n = 5}^{10} 20 - n$
Answer
Solution:
Step 1:
Transform the original summation to start from $n=1$ by decomposing it into two parts.
$$\sum_{n=5}^{10} (20 - n) = \sum_{n=1}^{10} (20 - n) - \sum_{n=1}^{4} (20 - n)$$
Step 2:
Compute the summation $\sum_{n=1}^{10} (20 - n)$.
Step 2.1:
Decompose the summation into two separate summations that are easier to evaluate.
$$\sum_{n=1}^{10} (20 - n) = \sum_{n=1}^{10} 20 - \sum_{n=1}^{10} n$$
Step 2.2:
Calculate $\sum_{n=1}^{10} 20$.
Step 2.2.1:
Use the formula for summing a constant value:
$$\sum_{k=1}^{n} c = cn$$
Step 2.2.2:
Insert the specific values into the formula.
$$(20)(10)$$
Step 2.2.3:
Perform the multiplication of $20$ by $10$.
$$200$$
Step 2.3:
Compute $\sum_{n=1}^{10} n$.
Step 2.3.1:
Apply the formula for the sum of the first $n$ natural numbers:
$$\sum_{k=1}^{n} k = \frac{n(n + 1)}{2}$$
Step 2.3.2:
Plug in the values and include the coefficient from the original expression.
$$(-1)\left(\frac{10(10 + 1)}{2}\right)$$
Step 2.3.3:
Simplify the expression.
Step 2.3.3.1:
Add $10$ and $1$ together.
$$-1\frac{10 \cdot 11}{2}$$
Step 2.3.3.2:
Multiply $10$ by $11$.
$$-1\left(\frac{110}{2}\right)$$
Step 2.3.3.3:
Divide $110$ by $2$.
$$-1 \cdot 55$$
Step 2.3.3.4:
Multiply $-1$ by $55$.
$$-55$$
Step 2.4:
Combine the results of the two summations.
$$200 - 55$$
Step 2.5:
Subtract $55$ from $200$.
$$145$$
Step 3:
Determine the summation $\sum_{n=1}^{4} (20 - n)$.
Step 3.1:
Write out the series for each value of $n$.
$$20 - 1 + 20 - 2 + 20 - 3 + 20 - 4$$
Step 3.2:
Simplify the series.
Step 3.2.1:
Subtract $1$ from $20$.
$$19 + 20 - 2 + 20 - 3 + 20 - 4$$
Step 3.2.2:
Subtract $2$ from $20$.
$$19 + 18 + 20 - 3 + 20 - 4$$
Step 3.2.3:
Combine $19$ and $18$.
$$37 + 20 - 3 + 20 - 4$$
Step 3.2.4:
Subtract $3$ from $20$.
$$37 + 17 + 20 - 4$$
Step 3.2.5:
Add $37$ and $17$.
$$54 + 20 - 4$$
Step 3.2.6:
Subtract $4$ from $20$.
$$54 + 16$$
Step 3.2.7:
Combine $54$ and $16$.
$$70$$
Step 4:
Substitute the computed summation values.
$$145 - 70$$
Step 5:
Subtract $70$ from $145$ to get the final result.
$$75$$
Knowledge Notes:
To solve the given problem, we utilized several key concepts in summation and arithmetic series:
Summation Properties: The ability to split a summation into the sum or difference of two or more summations, as long as the index of summation and the limits are the same.
Summation of a Constant: When summing a constant $c$ over $n$ terms, the result is simply $cn$.
Summation of the First $n$ Natural Numbers: The sum of the first $n$ natural numbers is given by the formula $\frac{n(n+1)}{2}$.
Arithmetic Operations: Basic arithmetic operations such as addition, subtraction, multiplication, and division are used to simplify expressions and compute the final result.
Decomposition of Complex Problems: Breaking down a complex problem into simpler, more manageable parts is a common strategy in mathematics. This approach makes it easier to apply known formulas and techniques to reach a solution.
By applying these concepts, we were able to evaluate the given summation by transforming it into simpler summations, calculating each part separately, and then combining the results to find the final answer.
