Question

Expand each sum. \(\sum_{k=1}^{n}(k+2)\)

Short answer

\( \sum_{k=1}^{n}(k+2) = \frac{n^2 + 5n}{2} \)

Step-by-step solution

Understand the Sum

The summation notation \(\sum_{k=1}^{n}(k+2)\) indicates that we need to sum the expression \(k+2\) for all integer values of \k\ from 1 to \.

Expand the Sum

Write out the sum for a few terms to see the pattern: \(\begin{aligned} \sum_{k=1}^{n}(k+2) &= (1+2) + (2+2) + (3+2) + \ldots + (n+2) \end{aligned}\).

Separate the Sum

The sum can be separated into two distinct sums: \(\begin{aligned} \sum_{k=1}^{n}(k+2) &= \sum_{k=1}^{n} k + \sum_{k=1}^{n} 2 \end{aligned}\).

Evaluate the Separate Sums

Evaluate each separated sum individually: \(\begin{aligned} \sum_{k=1}^{n} k \) and \(\begin{aligned} \sum_{k=1}^{n} 2 \end{aligned}\).

Find the Sum of k

Recall that the sum of the first \(n\) positive integers is given by the formula \[ \sum_{k=1}^{n} k = \frac{n(n+1)}{2} \].

Find the Sum of 2

The sum of the constant 2 for \(n\) terms is \(\begin{aligned} \sum_{k=1}^{n} 2 = 2n \end{aligned}\).

Combine the Results

Add the results of the two sums together: \(\begin{aligned} \sum_{k=1}^{n} (k+2) = \frac{n(n+1)}{2} + 2n \end{aligned}\).

Simplify

Factor out \(n\) from the expression: \(\begin{aligned} \sum_{k=1}^{n} (k+2) &= \frac{n(n+1)}{2} + 2n \) \(\begin{aligned} &= \frac{n(n+1) + 4n}{2} \) \(\begin{aligned} &= \frac{n^2 + n + 4n}{2} \) \(\begin{aligned} &= \frac{n^2 + 5n}{2} \end{aligned}\).

Key concepts

properties of summations

Summation notation is a compact way to represent the sum of a sequence of terms. The expression \(\begin{aligned} \sum_{k=1}^{n}(k+2) \end{aligned}\) means that we sum the expression \(k+2\) from \(k=1\) to \. Understanding the properties of summations helps simplify complex expressions. Here are some key properties:

• Linearity: If you have a sum of terms, you can split the sum into two separate sums. For example, \(\begin{aligned} \sum_{k=1}^{n}(a_k + b_k) = \sum_{k=1}^{n} a_k + \sum_{k=1}^{n} b_k \end{aligned}\).
• Constant multiples: Multiplying a term inside the summation by a constant factor can be taken outside the summation. For instance, \(\begin{aligned} \sum_{k=1}^{n} c a_k = c \sum_{k=1}^{n} a_k \end{aligned}\).

These properties are used to break down complex summations into simpler parts which are easier to manage.

arithmetic series

An arithmetic series is the sum of the terms in an arithmetic sequence. An arithmetic sequence has a common difference between each term. For example, in the sequence 1, 3, 5, 7,..., the difference between consecutive terms is 2.

• General form: The n-th term of an arithmetic sequence can be found using the formula: \(a_n = a_1 + (n-1)d\), where \a_1\ is the first term and \d\ is the common difference.
• Sum of an arithmetic series: The sum of the first \ terms of an arithmetic series can be calculated using the formula: \[ S_n = \frac{n}{2} \left(2a_1 + (n-1)d\right) \], which simplifies to \[ S_n = \frac{n(a_1 + a_n)}{2} \], where \a_n\ is the n-th term.

Understanding these concepts makes it easier to solve summation problems involving arithmetic series.

sum of integers formula

One of the most frequently used formulas in summation problems is the sum of the first \ positive integers. The formula is:

\[ \sum_{k=1}^{n} k = \frac{n(n+1)}{2} \]

This formula can be derived by pairing terms from opposite ends of the sequence and noticing that each pair sums to the same value. For example, in the sequence 1, 2, 3, ..., n:

• Pairing terms: \(1 + n\), \(2 + (n-1)\), \(3 + (n-2)\), and so on. Each pair sums to \+1\.
• Number of pairs: There are \frac{n}{2}\ pairs.

Multiplying the sum of each pair by the number of pairs gives us the formula for the sum of the first \ integers.

constant term summation

Summing a constant term repeatedly is straightforward. If you sum the same number \(c\) over \ terms, the result is simply the constant multiplied by the number of terms. For example:

• \[ \sum_{k=1}^{n} c = c \cdot n \]

In our example, the constant term is 2. Thus, summing 2 over \ terms gives \[ \sum_{k=1}^{n} 2 = 2n \]

This simple property is often used alongside other summation properties to break down and simplify more complex summations.