Evaluate the Summation sum from k=1 to 10 of 3k+7

Solution:

Step 1

Decompose the given summation into two separate summations that are easier to handle. $\sum_{k = 1}^{10} (3k + 7) = 3\sum_{k = 1}^{10} k + \sum_{k = 1}^{10} 7$

Step 2

Calculate $3 \sum_{k = 1}^{10} k$.

Step 2.1

Use 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, remembering to include the coefficient: $3\left(\frac{10(10 + 1)}{2}\right)$

Step 2.3

Proceed with the simplification.

Step 2.3.1

Combine $10$ and $1$: $3\frac{10 \cdot 11}{2}$

Step 2.3.2

Calculate $10 \times 11$: $3\left(\frac{110}{2}\right)$

Step 2.3.3

Divide $110$ by $2$: $3 \cdot 55$

Step 2.3.4

Multiply $3$ by $55$: $165$

Step 3

Calculate $\sum_{k = 1}^{10} 7$.

Step 3.1

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

Step 3.2

Plug in the given values: $(7)(10)$

Step 3.3

Perform the multiplication: $70$

Step 4

Combine the results from the two summations: $165 + 70$

Step 5

Finish by adding $165$ and $70$: $235$

Knowledge Notes:

The problem involves evaluating a finite arithmetic series, which is a summation of terms in a sequence where each term is generated by adding a constant difference to the previous term. The given series is a linear polynomial in $k$, and the summation can be split into two parts: one that is linear in $k$ and one that is a constant.

1. Summation of a Linear Polynomial: The sum of the first $n$ natural numbers is given by the formula $\sum_{k = 1}^{n} k = \frac{n(n + 1)}{2}$. This is an arithmetic series with a common difference of $1$.

2. Summation of a Constant: The sum of a constant $c$ over $n$ terms is simply $c$ multiplied by the number of terms, $n$, which is expressed as $\sum_{k = 1}^{n} c = cn$.

3. Distributive Property of Summation: The summation operator is linear, which means that $\sum_{k = 1}^{n} (a_k + b_k) = \sum_{k = 1}^{n} a_k + \sum_{k = 1}^{n} b_k$, where $a_k$ and $b_k$ are sequences dependent on $k$. This property allows us to split a complex summation into simpler parts that can be evaluated using known formulas.

4. Arithmetic Operations: Basic arithmetic operations such as addition, multiplication, and division are used to simplify expressions within the summation process.

By understanding these concepts, one can systematically approach the evaluation of summations and apply these principles to solve similar problems.