Evaluate the Summation sum from i=1 to 6 of 8i+3
The question provided is asking for the calculation of a specific mathematical summation. This summation requires adding together multiple terms, each defined by a formula, as the index variable "i" varies from the initial value of 1 to the final value of 6. The formula to generate each term of the summation is "8i + 3," where "i" represents the current value of the index within the given range. The task is to work out the sum of all these terms generated as "i" progresses from 1 to 6.
$\sum_{i = 1}^{6} 8 i + 3$
Answer
Solution:
Step 1: Decompose the given summation
Separate the original summation into two distinct summations that are easier to manage.
$$\sum_{i = 1}^{6} (8i + 3) = 8\sum_{i = 1}^{6} i + \sum_{i = 1}^{6} 3$$
Step 2: Calculate the first summation $8 \sum_{i = 1}^{6} i$
Step 2.1: Apply the arithmetic series formula
Use the formula for the sum of the first $n$ natural numbers:
$$\sum_{i = 1}^{n} i = \frac{n(n + 1)}{2}$$
Step 2.2: Plug in the upper limit of the summation and include the coefficient
Insert the value of $n$ into the formula and multiply by $8$:
$$8 \left( \frac{6(6 + 1)}{2} \right)$$
Step 2.3: Simplify the expression
Step 2.3.1: Perform the addition inside the parentheses
Calculate the sum within the numerator:
$$8 \frac{6 \cdot 7}{2}$$
Step 2.3.2: Simplify the fraction by reducing common factors
Reduce the fraction by dividing both the numerator and the coefficient by $2$:
$$4 \cdot \frac{42}{1}$$
Step 2.3.3: Multiply the remaining terms
Complete the multiplication to find the sum:
$$4 \cdot 42 = 168$$
Step 3: Calculate the second summation $\sum_{i = 1}^{6} 3$
Step 3.1: Use the formula for the sum of a constant
The sum of a constant $c$ repeated $n$ times is:
$$\sum_{i = 1}^{n} c = c \cdot n$$
Step 3.2: Insert the values into the formula
Substitute the constant $3$ and the number of terms $6$:
$$3 \cdot 6$$
Step 3.3: Perform the multiplication
Find the product to get the sum of the constants:
$$3 \cdot 6 = 18$$
Step 4: Combine the results of the two summations
Add the results from Step 2 and Step 3:
$$168 + 18$$
Step 5: Calculate the final sum
Add the two sums to get the final result:
$$168 + 18 = 186$$
Knowledge Notes:
The problem involves evaluating a finite arithmetic series, which is a sequence of numbers with a common difference between consecutive terms. In this case, the series is a linear function of $i$, with each term being of the form $8i + 3$. The problem-solving process involves the following knowledge points:
Summation Notation: The sigma notation $\sum$ is used to denote the sum of a sequence of numbers. The expression under the sigma symbol indicates the general form of the terms in the sequence.
Arithmetic Series Formula: The sum of the first $n$ natural numbers is given by the formula:
$$\sum_{i = 1}^{n} i = \frac{n(n + 1)}{2}$$ This formula is derived from the observation that the sum of the first and the last term, the second and the second-to-last term, and so on, are all equal to $n + 1$.
Summation of a Constant: The sum of a constant $c$ repeated $n$ times is simply $c$ multiplied by $n$:
$$\sum_{i = 1}^{n} c = c \cdot n$$
Distributive Property: The summation operation is linear, which allows us to distribute it over addition and separate the summation into the sum of two simpler summations.
Simplification of Fractions: When simplifying expressions involving fractions, it is often useful to cancel out common factors to make the calculation easier.
By applying these principles, the problem is broken down into simpler parts that can be solved step by step, leading to the final result.
