Problem 21 - Thomas Calculus

Question

Evaluate the sums in Exercises \(19-28\). $$ \sum_{k=1}^{7}(-2 k) $$

Step-by-Step Solution

Answer

The sum is -56.

Step 1: Understand the Summation Notation

The given problem is a summation, written in the format \( \sum_{k=1}^{7}(-2k) \). This notation means that we need to calculate the sum of all the terms \(-2k\) where \(k\) ranges from \(1\) to \(7\).

Step 2: Write the Individual Terms

Substitute the values of \(k\) from \(1\) to \(7\) into the expression \(-2k\) to find each individual term. These are: \(-2(1)\), \(-2(2)\), \(-2(3)\), \(-2(4)\), \(-2(5)\), \(-2(6)\), \(-2(7)\).

Step 3: Calculate Each Term

Compute each expression: \(-2(1) = -2\),\(-2(2) = -4\),\(-2(3) = -6\),\(-2(4) = -8\),\(-2(5) = -10\),\(-2(6) = -12\),\(-2(7) = -14\).

Step 4: Add All Terms Together

Now, add all the calculated terms together:\(-2 + (-4) + (-6) + (-8) + (-10) + (-12) + (-14)\).

Step 5: Calculate the Total Sum

Perform the addition step-by-step:- First, \(-2 + (-4) = -6\)- Then, \(-6 + (-6) = -12\)- Next, \(-12 + (-8) = -20\)- Continuing, \(-20 + (-10) = -30\)- Then, \(-30 + (-12) = -42\)- Finally, \(-42 + (-14) = -56\)Therefore, the sum is \(-56\).

Key Concepts

Arithmetic Series

In the world of mathematics, an arithmetic series is a crucial concept. It is a series of numbers in which the difference between consecutive terms is constant. This difference is commonly referred to as the "common difference." For example, in the series \(2, 4, 6, 8, ...\), each term increases by a common difference of \(2\). This structure allows mathematicians and students to predict and calculate unknown terms using simple formulas.

There is a straightforward formula used to find the sum of an arithmetic series: if \(a\) is the first term, \(d\) is the common difference, and \(n\) is the number of terms, the sum \(S_n\) is given by \(S_n = \frac{n}{2} (2a + (n-1)d)\). Understanding this formula can simplify the process of calculating the sum of arithmetic series. Use these rules and formulas whenever you need to handle arithmetic sequences.

Sequence

A sequence is essentially a list of numbers arranged in a specific order. It can be finite or infinite, depending on how many numbers it contains. Sequences are fundamental to studying and understanding patterns within mathematics. Each number in a sequence is known as a "term."

In our context, the sequence \(-2(1), -2(2), ..., -2(7)\) represents a sequence with a clear pattern. Each term is generated by multiplying \(k\) (a natural number starting from 1) by \(-2\). This gives us terms like \(-2, -4, -6\), each following a rule or pattern. Recognizing sequences helps in understanding how to sum such numbers efficiently, especially when dealing with a large set of terms.

Series Calculation

Series calculation involves summing the terms of a sequence. When we symbolize series using sigma notation \(\sum\), it allows us to sum numerous terms in a compact form. This is especially useful when dealing with long sequences.

For instance, in the series \(\sum_{k=1}^{7} (-2k)\), we are tasked to calculate the sum of each term created by replacing \(k\) with each integer from 1 to 7. Identifying each term's calculation, as done in our problem \((-2 \times 1 = -2\), etc.), and then summing them up, leads us to discover the overall series sum. Such practices are invaluable in mathematics for simplifying and managing large sums.

Finite Sum

A finite sum pertains to adding up a certain fixed count of numbers. Unlike an infinite series, which continues indefinitely, a finite sum has a defined endpoint. In arithmetic, finite sums appear frequently, especially in problems involving sequences with a specified range.

Consider the example of the finite sum \(-2 + (-4) + (-6) + (-8) + (-10) + (-12) + (-14) = -56\). Here, only the first seven terms are added together, as dictated by the range \(k=1\) to \(k=7\). This clearly illustrates how finite sums neatly conclude after a set number of terms, a concept that provides straightforward solutions to bound problems efficiently. Employ these finite sums when tackling questions where only part of an infinite series is needed.