Evaluate the Summation sum from i=1 to 8 of 3i-1
The problem presented is asking for the evaluation of a finite arithmetic series. Specifically, it is asking you to calculate the sum of a sequence of numbers generated by the given expression "3i-1" as the variable 'i' takes on integer values from 1 to 8. Each term of the sequence is determined by multiplying the index 'i' by 3 and then subtracting 1. Once you form this sequence by applying the expression to each value of 'i' within the specified range, you are expected to add all the terms together to produce a single numerical result, which is the summation of the series.
$\sum_{i = 1}^{8} 3 i - 1$
Answer
Solution:
Step:1
Decompose the given summation into two separate summations that adhere to the properties of summation. $\sum_{i = 1}^{8} (3i - 1) = 3\sum_{i = 1}^{8} i - \sum_{i = 1}^{8} 1$
Step:2
Calculate $3 \sum_{i = 1}^{8} i$.
Step:2.1
Utilize the formula for the sum of the first $n$ natural numbers: $\sum_{i = 1}^{n} i = \frac{n(n + 1)}{2}$
Step:2.2
Insert the upper limit of the summation into the formula and include the coefficient: $3 \left(\frac{8(8 + 1)}{2}\right)$
Step:2.3
Proceed with simplification.
Step:2.3.1
Combine $8$ and $1$: $3 \frac{8 \cdot 9}{2}$
Step:2.3.2
Perform the multiplication of $8$ and $9$: $3 \left(\frac{72}{2}\right)$
Step:2.3.3
Divide $72$ by $2$: $3 \cdot 36$
Step:2.3.4
Multiply $3$ by $36$: $108$
Step:3
Compute $\sum_{i = 1}^{8} -1$.
Step:3.1
Apply the formula for the summation of a constant term: $\sum_{i = 1}^{n} c = cn$
Step:3.2
Replace the constant and upper limit in the formula: $(-1) \cdot 8$
Step:3.3
Calculate the product of $-1$ and $8$: $-8$
Step:4
Combine the results from the two summations: $108 - 8$
Step:5
Finalize by subtracting $8$ from $108$: $100$
Knowledge Notes:
The problem involves evaluating a summation expression, which is a common task in algebra and calculus. The process of solving such a problem typically involves the following knowledge points:
Summation Properties: Summation expressions can often be broken down into simpler parts using properties such as linearity, which allows us to split a summation of terms into separate summations.
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 is a specific case of an arithmetic series where the common difference is $1$.
Summation of a Constant: The sum of a constant $c$ over $n$ terms is simply $cn$, because each term in the summation is the same.
Algebraic Simplification: After applying the formulas, the remaining task is to simplify the algebraic expression by performing basic arithmetic operations such as addition, multiplication, and division.
Combining Results: Once individual summations are evaluated, their results are combined (added, in this case) to obtain the final answer.
Understanding and applying these concepts allows one to systematically approach and solve summation problems.
