Evaluate the Summation sum from k=1 to n of k+6
The question is asking for the evaluation of a mathematical summation. Specifically, you are to find the total value obtained by adding up the terms in the sequence that follow the pattern "k+6" for each integer value of k, starting from k=1 and continuing up to and including k=n, where n is some variable or specified upper limit. This involves using the rules and formulas for summation that deal with arithmetic sequences or applying properties of summations to simplify and calculate the sum of all these terms.
$\sum_{k = 1}^{n} k + 6$
Answer
Solution:
Step 1:
We begin by interpreting the given summation expression. The task is to find the sum of all terms in the sequence where each term is defined as the sum of an integer $k$ and the number 6, with $k$ starting at 1 and increasing by 1 until it reaches $n$.
Step 2:
We can separate the summation into two parts: the summation of $k$ from 1 to $n$, and the summation of the constant 6 from 1 to $n$.
$$\sum_{k = 1}^{n} (k + 6) = \sum_{k = 1}^{n} k + \sum_{k = 1}^{n} 6$$
Step 3:
The summation of $k$ from 1 to $n$ is a well-known arithmetic series whose sum can be calculated using the formula:
$$\sum_{k = 1}^{n} k = \frac{n(n + 1)}{2}$$
Step 4:
The summation of the constant 6 from 1 to $n$ is simply $n$ times 6, because we are adding the number 6 exactly $n$ times:
$$\sum_{k = 1}^{n} 6 = 6n$$
Step 5:
Now, we combine the results from Step 3 and Step 4 to find the total sum:
$$\sum_{k = 1}^{n} (k + 6) = \frac{n(n + 1)}{2} + 6n$$
Step 6:
Simplify the expression to get the final result:
$$\sum_{k = 1}^{n} (k + 6) = \frac{n^2 + n}{2} + \frac{12n}{2} = \frac{n^2 + n + 12n}{2} = \frac{n^2 + 13n}{2}$$
Thus, the sum of the series is $\frac{n^2 + 13n}{2}$.
Knowledge Notes:
Summation Notation: Summation notation is a convenient way to express the sum of a sequence of terms. The symbol $\sum$ represents the sum, and the expression below and above the symbol indicates the range of indices over which to sum the terms.
Arithmetic Series: An arithmetic series is the sum of the terms of an arithmetic sequence, a sequence of numbers with a constant difference between consecutive terms. The sum of the first $n$ terms of an arithmetic sequence can be found using the formula $\sum_{k = 1}^{n} k = \frac{n(n + 1)}{2}$.
Summation of a Constant: When summing a constant $c$ over $n$ terms, the result is simply $n$ times the constant: $\sum_{k = 1}^{n} c = nc$.
Combining Sums: When dealing with summations, sums can be separated or combined as long as the range of summation is the same for each sum. This property allows us to break down more complex summations into simpler parts that can be calculated individually and then combined for the final result.
Simplifying Expressions: After finding the individual sums, the expressions can be combined and simplified using algebraic manipulation to reach the final, simplified form of the sum.
