Evaluate the Summation sum from x=1 to 4 of 20-x^2
The problem is asking you to calculate the result of a finite summation. Specifically, you are to find the sum of the expression "20 - x^2" as the variable x takes on integer values from 1 to 4, inclusive. You are expected to plug in each integer value for x into the expression, calculate the result for each one, and then add all those results together to get the final sum.
$\sum_{x = 1}^{4} 20 - x^{2}$
Answer
Solution:
Step 1:
Write out the summation term for each value of $x$: $20 - 1^2, 20 - 2^2, 20 - 3^2, 20 - 4^2$.
Step 2:
Perform the simplification process.
Step 2.1:
Calculate $1^2$: $20 - 1 \cdot 1 + 20 - 2^2 + 20 - 3^2 + 20 - 4^2$.
Step 2.2:
Evaluate the product of $-1$ and $1$: $20 - 1 + 20 - 2^2 + 20 - 3^2 + 20 - 4^2$.
Step 2.3:
Subtract $1$ from $20$: $19 + 20 - 2^2 + 20 - 3^2 + 20 - 4^2$.
Step 2.4:
Calculate $2^2$: $19 + 20 - 4 + 20 - 3^2 + 20 - 4^2$.
Step 2.5:
Evaluate the product of $-1$ and $4$: $19 + 20 - 4 + 20 - 3^2 + 20 - 4^2$.
Step 2.6:
Subtract $4$ from $20$: $19 + 16 + 20 - 3^2 + 20 - 4^2$.
Step 2.7:
Combine $19$ and $16$: $35 + 20 - 3^2 + 20 - 4^2$.
Step 2.8:
Calculate $3^2$: $35 + 20 - 9 + 20 - 4^2$.
Step 2.9:
Evaluate the product of $-1$ and $9$: $35 + 20 - 9 + 20 - 4^2$.
Step 2.10:
Subtract $9$ from $20$: $35 + 11 + 20 - 4^2$.
Step 2.11:
Combine $35$ and $11$: $46 + 20 - 4^2$.
Step 2.12:
Calculate $4^2$: $46 + 20 - 16$.
Step 2.13:
Evaluate the product of $-1$ and $16$: $46 + 20 - 16$.
Step 2.14:
Subtract $16$ from $20$: $46 + 4$.
Step 2.15:
Combine $46$ and $4$: $50$.
The final result is $50$.
Knowledge Notes:
The problem involves evaluating a summation, which is a process of adding up a sequence of terms. The summation given is $\sum_{x=1}^{4} (20 - x^2)$, which means we need to calculate the expression $20 - x^2$ for each integer value of $x$ from $1$ to $4$ and then sum these values.
Key knowledge points include:
Understanding of summation notation ($\Sigma$), which is a concise way to represent the sum of a sequence of terms.
Basic arithmetic operations: addition, subtraction, multiplication, and exponentiation.
Squaring a number, which means multiplying the number by itself (e.g., $2^2 = 2 \times 2 = 4$).
The properties of negative numbers, particularly how multiplying by $-1$ changes the sign of a number.
The importance of following the order of operations in arithmetic, which is commonly abbreviated as PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).
In this problem, each step of the solution involves applying arithmetic operations to simplify the expression for each term in the summation, and then adding the results to find the total sum. The process is systematic, ensuring that each term is evaluated and simplified before moving on to the next.
