Evaluate the Summation sum from 1 to 10 of 10+3(n-1)
The question asks for an evaluation of a mathematical summation. Specifically, the summation is defined by the function 10 + 3(n - 1), in which the variable 'n' will take on integer values starting from 1 and increasing to 10. To solve this, one would have to calculate the value of this function for each integer value of 'n' within the limits of the summation (from 1 to 10), and then add up all of these values to get the total sum.
$\sum_{1}^{10} 10 + 3 \left(\right. n - 1 \left.\right)$
Answer
Solution:
Step 1: Simplify the given summation expression.
Step 1.1: Break down the expression term by term.
Step 1.1.1: Use the distributive property to expand: $10 + 3(n - 1)$.
Step 1.1.2: Perform the multiplication: $3 \times (-1)$ to get $10 + 3n - 3$.
Step 1.2: Combine like terms by subtracting $3$ from $10$ to obtain $3n + 7$.
Step 1.3: Represent the simplified summation as $\sum_{n=1}^{10} (3n + 7)$.
Step 2: Decompose the summation into two separate summations.
- $\sum_{n=1}^{10} (3n + 7) = 3\sum_{n=1}^{10} n + \sum_{n=1}^{10} 7$.
Step 3: Calculate the first summation $3\sum_{n=1}^{10} n$.
Step 3.1: Recall the formula for the sum of the first $n$ natural numbers: $\sum_{k=1}^{n} k = \frac{n(n + 1)}{2}$.
Step 3.2: Insert the upper limit of the summation into the formula and multiply by $3$: $3 \left(\frac{10(10 + 1)}{2}\right)$.
Step 3.3: Simplify the expression.
Step 3.3.1: Add $10$ and $1$ together.
Step 3.3.2: Calculate $10 \times 11$.
Step 3.3.3: Divide $110$ by $2$.
Step 3.3.4: Multiply $3$ by $55$ to get $165$.
Step 4: Evaluate the second summation $\sum_{n=1}^{10} 7$.
Step 4.1: Use the formula for the sum of a constant: $\sum_{k=1}^{n} c = cn$.
Step 4.2: Apply the formula with $c = 7$ and $n = 10$.
Step 4.3: Multiply $7$ by $10$ to get $70$.
Step 5: Combine the results of the two summations.
- Add the values from Step 3 and Step 4: $165 + 70$.
Step 6: Sum the final results to get the answer.
- The final sum is $235$.
Knowledge Notes:
To solve the given summation problem, we need to understand the following concepts:
Distributive Property: This property allows us to multiply a sum by multiplying each addend separately and then sum the products. For example, $a(b + c) = ab + ac$.
Summation Notation: Summation notation is a way to represent the sum of a sequence of terms. The symbol $\sum$ denotes summation, with the lower and upper limits of the summation provided below and above the symbol, respectively.
Arithmetic Series: The sum of the terms of an arithmetic sequence, where each term is a constant difference from the previous term. The sum of the first $n$ natural numbers is given by the formula $\sum_{k=1}^{n} k = \frac{n(n + 1)}{2}$.
Summation of a Constant: The sum of a constant $c$ repeated $n$ times is given by the formula $\sum_{k=1}^{n} c = cn$.
Combining Like Terms: This involves simplifying expressions by adding or subtracting terms that have the same variable raised to the same power.
Simplification of Expressions: This involves performing operations like addition, subtraction, multiplication, and division to reduce an expression to its simplest form.
By applying these concepts, we can evaluate the given summation by simplifying the expression, breaking it into parts that are easier to calculate, and then combining the results to find the final sum.
