Problem 6
Question
Find the common difference \(d\) for each arithmetic sequence. Do not use a calculator. $$t^{2}+q,-4 t^{2}+2 q,-9 t^{2}+3 q, \dots$$
Step-by-Step Solution
Verified Answer
The common difference is \(-5t^2 + q\).
1Step 1: Recognize the format of an arithmetic sequence
An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. This difference is known as the common difference, denoted by \(d\).
2Step 2: Identify two consecutive terms
Consider the first two terms of the sequence: \(t^2 + q\) and \(-4t^2 + 2q\). We will find the common difference \(d\) by subtracting the first term from the second term.
3Step 3: Subtract the first term from the second term
Calculate the difference between the second and the first term: \[(-4t^2 + 2q) - (t^2 + q)\]Simplify this expression to find:\[-4t^2 + 2q - t^2 - q = -5t^2 + q\]
4Step 4: Verify the common difference with the next terms
Verify the common difference by checking with the third term: \(-9t^2 + 3q\). Calculate the difference between the third term and the second term:\[(-9t^2 + 3q) - (-4t^2 + 2q) = -9t^2 + 3q + 4t^2 - 2q\]Simplify this expression to find:\[-5t^2 + q\]This confirms the common difference is consistent across the terms.
Key Concepts
Common DifferenceConsecutive TermsPatterns in Sequences
Common Difference
The common difference is a fundamental concept in arithmetic sequences. It represents the difference between any two consecutive terms in such a sequence. Essentially, it tells us how much each term is increasing or decreasing to give us the next term. For instance, if the common difference is 3, each following term will be 3 units greater than the previous term.
To find the common difference, simply pick any two consecutive terms from the sequence. Subtract the first term from the second term.
For example, in this sequence: \(t^2 + q, -4t^2 + 2q, -9t^2 + 3q\), you can calculate the common difference \(d\) as follows:
To find the common difference, simply pick any two consecutive terms from the sequence. Subtract the first term from the second term.
For example, in this sequence: \(t^2 + q, -4t^2 + 2q, -9t^2 + 3q\), you can calculate the common difference \(d\) as follows:
- Take the second term \(-4t^2 + 2q\).
- Subtract the first term \(t^2 + q\) from it to get: \((-4t^2 + 2q) - (t^2 + q) = -5t^2 + q\).
Consecutive Terms
Consecutive terms in an arithmetic sequence are terms that follow one another in the sequence's order. The nature of such sequences is defined by the regularity and predictability of the placement of these consecutive terms, thanks to the common difference.
When exploring consecutive terms in an arithmetic sequence, you simply take any term from the sequence and the term that immediately follows it. Each term is derived from the one before by adding the common difference.
For instance, consider the given sequence's first two terms, \(t^2 + q\) and \(-4t^2 + 2q\). The difference between these two consecutive terms illustrates the common difference.
Moreover, you can verify the arithmetic property by comparing any other pair of consecutive terms, like the second term \(-4t^2 + 2q\) and the third term \(-9t^2 + 3q\), and checking if their difference is consistent with the previously calculated common difference \(-5t^2 + q\). An arithmetic sequence assures this constancy in calculation.
When exploring consecutive terms in an arithmetic sequence, you simply take any term from the sequence and the term that immediately follows it. Each term is derived from the one before by adding the common difference.
For instance, consider the given sequence's first two terms, \(t^2 + q\) and \(-4t^2 + 2q\). The difference between these two consecutive terms illustrates the common difference.
Moreover, you can verify the arithmetic property by comparing any other pair of consecutive terms, like the second term \(-4t^2 + 2q\) and the third term \(-9t^2 + 3q\), and checking if their difference is consistent with the previously calculated common difference \(-5t^2 + q\). An arithmetic sequence assures this constancy in calculation.
Patterns in Sequences
Patterns in sequences, particularly arithmetic sequences, play a crucial role in understanding how sequences are structured. An arithmetic sequence pattern emerges clearly due to the consistency provided by the common difference. Every term in the sequence is modified by a fixed amount to yield the next one. This uniform increment or decrement describes a predictable pattern.
This pattern not only makes it simpler to understand and work with sequences but also forms the foundation between terms fortifying mathematical predictability in various applications.
- The arithmetic pattern is highlighted by a linear relationship, where plotting the sequence terms on a graph would produce a straight line.
- The regularity of arithmetic growth or decay reflects a steady pace of change.
This pattern not only makes it simpler to understand and work with sequences but also forms the foundation between terms fortifying mathematical predictability in various applications.
Other exercises in this chapter
Problem 6
Use mathematical induction to prove each statement. Assume that \(n\) is a positive integer. $$1+3+5+\dots+(2 n-1)=n^{2}$$
View solution Problem 6
State a sample space \(S\) with equally likely outcomes for each experiment. A die is rolled and then a coin is tossed.
View solution Problem 6
Evaluate each expression. Do not use a calculator. $$\frac{7 !}{6 !}$$
View solution Problem 6
Evaluate the following. In Exercises 17 and 18 , express the answer in terms of n. Do not use a calculator. $$\left(\begin{array}{l}7 \\\4\end{array}\right)$$
View solution