Evaluate the Summation sum from i=1 to 6 of 9i-2
The problem is asking to calculate the sum of a sequence of terms that follow the pattern given by the expression $9i - 2$. The variable $i$takes on integer values starting at 1 and ending at 6. For each value of $i$, you compute the expression $9i - 2$, and then you add up all of these computed values to get the final sum. This is a problem involving a summation operation, which is a fundamental concept in arithmetic and calculus.
$\sum_{i = 1}^{6} 9 i - 2$
Answer
Solution:
Step 1:
Decompose the given summation into two separate summations by distributing the summation operator over the terms within the summation. The original summation $\sum_{i = 1}^{6} (9i - 2)$ can be rewritten as $9\sum_{i = 1}^{6} i - 2\sum_{i = 1}^{6} 1$.
Step 2:
Calculate the first summation $9\sum_{i = 1}^{6} i$.
Step 2.1:
Use the arithmetic series sum formula for a sequence of 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: $9\left(\frac{6(6 + 1)}{2}\right)$.
Step 2.3:
Proceed with the simplification.
Step 2.3.1:
Compute the sum within the parentheses: $9\frac{6 \cdot 7}{2}$.
Step 2.3.2:
Perform the multiplication inside the fraction: $9\left(\frac{42}{2}\right)$.
Step 2.3.3:
Divide the fraction: $9 \cdot 21$.
Step 2.3.4:
Complete the multiplication: $189$.
Step 3:
Evaluate the second summation $-2\sum_{i = 1}^{6} 1$.
Step 3.1:
Apply the formula for the sum of a constant term: $\sum_{i = 1}^{n} c = cn$.
Step 3.2:
Place the given values into the formula: $(-2)(6)$.
Step 3.3:
Carry out the multiplication: $-12$.
Step 4:
Combine the results from the two summations: $189 - 12$.
Step 5:
Finalize the calculation by subtracting $12$ from $189$: $177$.
The returned result format is as follows:
Solution:"177"
Knowledge Notes:
The summation of a sequence of natural numbers from $1$ to $n$ is given by the formula $\sum_{i = 1}^{n} i = \frac{n(n + 1)}{2}$, which is derived from the properties of an arithmetic series.
The summation of a constant $c$ over $n$ terms is simply $cn$, as each term in the summation is the same.
When dealing with summations, it is often useful to break down complex expressions into simpler summations that can be evaluated using known formulas.
Distributive property of summation allows us to separate a summation of terms into individual summations, which can be handled independently.
Arithmetic operations within the summation (like addition, subtraction, and multiplication by a constant) can be performed step by step to simplify the expression before applying summation formulas.
