Evaluate the Summation sum from n=1 to 7 of n+15
The question asks for the evaluation of a mathematical expression that represents a summation. Specifically, you are to find the sum of a sequence of numbers where each number is derived from the formula n+15. The summation process should begin at n=1 and continue incrementally by 1 until n=7. Essentially, you would calculate the individual values for each n in the range from 1 to 7, add 15 to each of these values, and then sum all of these results together to get the final answer.
$\sum_{n = 1}^{7} n + 15$
Answer
Solution:
Step 1
Decompose the given summation into two separate summations that adhere to standard summation properties.
$$\sum_{n = 1}^{7} (n + 15) = \sum_{n = 1}^{7} n + \sum_{n = 1}^{7} 15$$
Step 2
Calculate the summation of the series of natural numbers up to 7, $\sum_{n = 1}^{7} n$.
Step 2.1
Utilize the arithmetic series summation formula:
$$\sum_{k = 1}^{n} k = \frac{n(n + 1)}{2}$$
Step 2.2
Insert the upper limit of the summation into the formula:
$$\frac{7(7 + 1)}{2}$$
Step 2.3
Proceed with the simplification process.
Step 2.3.1
Combine 7 and 1:
$$\frac{7 \cdot 8}{2}$$
Step 2.3.2
Perform the multiplication of 7 by 8:
$$\frac{56}{2}$$
Step 2.3.3
Complete the division of 56 by 2:
$$28$$
Step 3
Determine the summation of the constant 15 repeated 7 times, $\sum_{n = 1}^{7} 15$.
Step 3.1
Apply the formula for the summation of a constant value:
$$\sum_{k = 1}^{n} c = cn$$
Step 3.2
Place the constant and the number of terms into the formula:
$$(15)(7)$$
Step 3.3
Carry out the multiplication of 15 by 7:
$$105$$
Step 4
Combine the outcomes of the two summations:
$$28 + 105$$
Step 5
Sum the values of 28 and 105 to find the final result:
$$133$$
Knowledge Notes:
The process of evaluating a summation involves several key knowledge points:
Summation Properties: Summations can be split into the sum of individual summations when the terms are separable, as seen with $\sum (n + 15)$ being split into $\sum n + \sum 15$.
Arithmetic Series Summation Formula: The formula for the sum of the first $n$ natural numbers is $\sum_{k = 1}^{n} k = \frac{n(n + 1)}{2}$. This is derived from the fact that the sum of an arithmetic series is equal to the average of the first and last term, multiplied by the number of terms.
Summation of a Constant: The sum of a constant $c$ repeated $n$ times is given by $\sum_{k = 1}^{n} c = cn$. This is because each term in the summation is the same, so the sum is simply the constant multiplied by the number of terms.
Simplification: The process of simplifying expressions involves performing arithmetic operations such as addition, multiplication, and division in the correct order, following the rules of arithmetic.
By understanding these concepts, one can systematically approach and solve summation problems in a step-by-step manner, ensuring accuracy and efficiency in calculations.
