Evaluate the Summation sum from n=1 to 90 of 3n-7

Solution:

Step 1:

Decompose the given summation into two separate summations that are easier to manage using the distributive property of summation: $\sum_{n=1}^{90}(3n-7) = 3\sum_{n=1}^{90}n - \sum_{n=1}^{90}7$

Step 2:

Calculate the summation $3\sum_{n=1}^{90}n$.

Step 2.1:

Utilize the arithmetic series sum formula: $\sum_{k=1}^{n}k = \frac{n(n+1)}{2}$

Step 2.2:

Insert the upper limit of the summation into the formula and include the coefficient: $3\left(\frac{90(90+1)}{2}\right)$

Step 2.3:

Proceed with simplifying the expression.

Step 2.3.1:

Combine $90$ and $1$: $3\frac{90 \cdot 91}{2}$

Step 2.3.2:

Calculate the product of $90$ and $91$: $3\left(\frac{8190}{2}\right)$

Step 2.3.3:

Divide $8190$ by $2$: $3 \cdot 4095$

Step 2.3.4:

Multiply $3$ by $4095$: $12285$

Step 3:

Compute the summation $\sum_{n=1}^{90}-7$.

Step 3.1:

Apply the constant series sum formula: $\sum_{k=1}^{n}c = cn$

Step 3.2:

Replace the constant and the upper limit of the summation: $(-7)(90)$

Step 3.3:

Perform the multiplication of $-7$ and $90$: $-630$

Step 4:

Combine the outcomes of the two summations: $12285 - 630$

Step 5:

Finalize by subtracting $630$ from $12285$: $11655$

Solution:"11655"

Knowledge Notes:

1. Summation Properties: Summation (or sigma notation) allows for the concise expression of the sum of a sequence of values. The distributive property lets us break down a complex summation into simpler components, often separating variable terms from constants.

2. Arithmetic Series: An arithmetic series is the sum of the terms of an arithmetic sequence, a sequence in which each term after the first is obtained by adding a constant difference to the preceding term. The sum of the first $n$ natural numbers is given by the formula $\sum_{k=1}^{n}k = \frac{n(n+1)}{2}$.

3. Sum of a Constant Sequence: The sum of a constant $c$ over $n$ terms is simply $cn$, because each term in the sequence is the same and there are $n$ such terms.

4. Simplification: Simplifying expressions involves performing arithmetic operations like addition, multiplication, and division to reduce an expression to its simplest form.

5. Combining Results: After evaluating separate parts of a problem, the final step often involves combining these parts to obtain the final result, which may require addition, subtraction, or other operations.