Evaluate the Summation sum from x=51 to 100 of 7x
The question is asking for an explanation of a mathematical operation known as a summation. Specifically, the instruction is to calculate the total sum of a series of terms that follow the general form of 7 times a variable x. The variable x is to take on integer values starting at 51 and ending at 100. In other words, you will need to compute the sum of 7 multiplied by each integer from 51 up to and including 100. The process involves adding up all these individual products to find the final result. The question does not require an actual calculation or answer, only an explanation of what the problem is asking for.
$\sum_{x = 51}^{100} 7 x$
Answer
Solution:
Step 1:
Adjust the range of the summation to start from $x=1$.
$$\sum_{x=51}^{100} 7x = \sum_{x=1}^{100} 7x - \sum_{x=1}^{50} 7x$$
Step 2:
Calculate the first summation $\sum_{x=1}^{100} 7x$.
Step 2.1:
Extract the constant $7$ from the summation.
$$7 \sum_{x=1}^{100} x$$
Step 2.2:
Use the arithmetic series sum formula.
$$\sum_{k=1}^{n} k = \frac{n(n+1)}{2}$$
Step 2.3:
Plug in $n=100$ and multiply by $7$.
$$7 \left( \frac{100(100+1)}{2} \right)$$
Step 2.4:
Carry out the simplification.
Step 2.4.1:
Combine $100$ and $1$.
$$7 \frac{100 \cdot 101}{2}$$
Step 2.4.2:
Perform the multiplication of $100$ and $101$.
$$7 \left( \frac{10100}{2} \right)$$
Step 2.4.3:
Divide $10100$ by $2$.
$$7 \cdot 5050$$
Step 2.4.4:
Multiply $7$ by $5050$.
$$35350$$
Step 3:
Calculate the second summation $\sum_{x=1}^{50} 7x$.
Step 3.1:
Extract the constant $7$ from the summation.
$$7 \sum_{x=1}^{50} x$$
Step 3.2:
Use the arithmetic series sum formula.
$$\sum_{k=1}^{n} k = \frac{n(n+1)}{2}$$
Step 3.3:
Plug in $n=50$ and multiply by $7$.
$$7 \left( \frac{50(50+1)}{2} \right)$$
Step 3.4:
Carry out the simplification.
Step 3.4.1:
Combine $50$ and $1$.
$$7 \frac{50 \cdot 51}{2}$$
Step 3.4.2:
Perform the multiplication of $50$ and $51$.
$$7 \left( \frac{2550}{2} \right)$$
Step 3.4.3:
Divide $2550$ by $2$.
$$7 \cdot 1275$$
Step 3.4.4:
Multiply $7$ by $1275$.
$$8925$$
Step 4:
Substitute the computed summations.
$$35350 - 8925$$
Step 5:
Execute the subtraction to find the final result.
$$26425$$
Knowledge Notes:
The problem involves evaluating a summation of a linear sequence. Here are the relevant knowledge points:
Summation Notation: Summation notation (also known as sigma notation) is a way to write the sum of a sequence of terms. It is represented by the Greek letter sigma ($\Sigma$) followed by an expression that varies according to an index variable.
Arithmetic Series: An arithmetic series is the sum of the terms of an arithmetic sequence, a sequence in which each term after the first is obtained by adding a constant difference to the previous term. The sum of the first $n$ terms of an arithmetic sequence can be found using the formula:
$$\sum_{k=1}^{n} k = \frac{n(n+1)}{2}$$
Factoring Constants from Summations: A constant multiplier within the sum can be factored out of the summation, simplifying the expression. For example:
$$\sum_{k=1}^{n} ck = c\sum_{k=1}^{n} k$$ where $c$ is a constant.
Adjusting the Index of Summation: Sometimes it's necessary to adjust the index of summation to simplify the problem. This can involve splitting the summation into two parts or changing the lower and upper bounds of the summation index.
Basic Algebraic Manipulation: The process involves basic algebraic steps such as distributing multiplication over addition, simplifying fractions, and performing arithmetic operations.
By understanding and applying these concepts, one can solve summation problems involving linear sequences.
