Evaluate the Summation sum from i=1 to 6 of 9i-2

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.