Evaluate the Summation sum from k=1 to 20 of 5k+3

Solution:

Step 1: Decompose the given summation

Break down the given summation into two separate summations that are easier to handle: $\sum_{k=1}^{20}(5k+3) = 5\sum_{k=1}^{20}k + \sum_{k=1}^{20}3$

Step 2: Evaluate the first part of the summation $5\sum_{k=1}^{20}k$

Step 2.1: Apply the arithmetic series formula

Use the formula for the sum of the first $n$ natural numbers: $\sum_{k=1}^{n}k = \frac{n(n+1)}{2}$

Step 2.2: Insert the upper limit of the summation into the formula

Calculate the sum by plugging in $n=20$: $5\left(\frac{20(20+1)}{2}\right)$

Step 2.3: Simplify the expression

Step 2.3.1: Perform the addition inside the parentheses

Calculate $20+1$: $5\frac{20\cdot21}{2}$

Step 2.3.2: Carry out the multiplication

Multiply $20$ by $21$: $5\left(\frac{420}{2}\right)$

Step 2.3.3: Execute the division

Divide $420$ by $2$: $5\cdot210$

Step 2.3.4: Finalize the multiplication

Multiply $5$ by $210$: $1050$

Step 3: Evaluate the second part of the summation $\sum_{k=1}^{20}3$

Step 3.1: Use the constant series formula

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

Step 3.2: Substitute the values into the formula

Calculate the sum with $c=3$ and $n=20$: $(3)(20)$

Step 3.3: Perform the multiplication

Multiply $3$ by $20$: $60$

Step 4: Combine the results of the two summations

Add the results from step 2 and step 3: $1050 + 60$

Step 5: Calculate the final sum

Add $1050$ and $60$ to get the final result: $1110$

Knowledge Notes:

The problem involves evaluating a summation of an arithmetic sequence. The key knowledge points to understand and solve this problem are:

1. 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 pattern of the sequence, and the limits of summation indicate the starting and ending indices.

2. Arithmetic Series: An arithmetic series is the sum of the terms of an arithmetic sequence, which is a sequence of numbers with a constant difference between consecutive terms.

3. Sum of the First n Natural Numbers: The sum of the first $n$ natural numbers is given by the formula $\sum_{k=1}^{n}k = \frac{n(n+1)}{2}$. This formula is derived from the properties of arithmetic series.

4. Sum of a Constant Series: The sum of a constant series (where each term is the same) is simply the constant multiplied by the number of terms: $\sum_{k=1}^{n}c = cn$.

5. Distributive Property: This property is used to split the original summation into two separate summations, allowing us to apply the formulas for arithmetic and constant series.

6. Basic Algebraic Operations: Simplifying the expressions within the summation involves basic algebraic operations such as addition, multiplication, and division.

Understanding these concepts is crucial for solving summation problems efficiently and correctly.