Problem 57 - Algebra 2

Question

CHALLENGE Determine whether each statement is true or false. If true, explain. If false, provide a counterexample. There is no sequence that is both arithmetic and geometric.

Step-by-Step Solution

Answer

The statement is false; a constant sequence is both arithmetic and geometric.

Step 1: Define Arithmetic and Geometric Sequences

An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. This difference is called the common difference, denoted by \(d\). A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio, denoted by \(r\).

Step 2: Analyze the Statement

The statement claims there is no sequence that is both arithmetic (with a common difference) and geometric (with a common ratio).

Step 3: Special Case: Constant Sequence

Consider a sequence where all terms are the same, e.g., \(a, a, a, \ldots\). This sequence is arithmetic because each term fits \(a + d = a\), so the common difference \(d = 0\). It is also geometric because each term fits \(ar = a\), so the common ratio \(r = 1\).

Step 4: Conclusion

A constant sequence is both arithmetic (with zero common difference) and geometric (with common ratio of one). Thus, the statement "There is no sequence that is both arithmetic and geometric" is false.

Key Concepts

Arithmetic Sequence

An arithmetic sequence is a series of numbers where each term after the first is found by adding a constant, known as the common difference, to the previous term. To visualize this, imagine starting with the first term, let's say 3. If the common difference is 2, then the subsequent terms follow as 3, 5, 7, 9, and so on.

• First term: 3
• Second term: 3 + 2 = 5
• Third term: 5 + 2 = 7
• Fourth term: 7 + 2 = 9

One key feature of arithmetic sequences is that they follow a linear pattern. This means if you were to graph the terms, they would form a straight line. The formula for the nth term of an arithmetic sequence is given by: \[ a_n = a_1 + (n-1) imes d \] where:
- \( a_n \) is the nth term, - \( a_1 \) is the first term, and - \( d \) is the common difference.

Geometric Sequence

A geometric sequence, on the other hand, is made up of numbers in which each term is derived by multiplying the previous term by a constant value. This constant is called the common ratio. If our first term is 2 and the common ratio is 3, then the sequence would appear as 2, 6, 18, 54, etc.

• First term: 2
• Second term: 2 \( \times \) 3 = 6
• Third term: 6 \( \times \) 3 = 18
• Fourth term: 18 \( \times \) 3 = 54

Geometric sequences are exponential in nature. This gives them a curve-like pattern when graphed, instead of a straight line. The nth term of a geometric sequence can be calculated with the formula: \[ a_n = a_1 \times r^{(n-1)} \] where:
- \( a_n \) is the nth term, - \( a_1 \) is the first term, and - \( r \) is the common ratio.

Common Difference

The common difference in an arithmetic sequence is the fixed number that we add to each term to get the next term. It's like a constant increment that shapes the sequence. Using the arithmetic sequence example from earlier, where the sequence is 3, 5, 7, 9, 11, the common difference is 2.

• Subtract the first term from the second: 5 - 3 = 2
• Continue this for the next terms: 7 - 5 = 2, and so forth

This constant difference, indicated by \(d\), is crucial because it determines the speed at which the sequence progresses. A larger common difference means the numbers in the sequence spread out more quickly, whereas a smaller one implies they will be closer together.

Common Ratio

A geometric sequence relies on a common ratio for its growth or decline from one term to the next. The common ratio is the fixed number by which we multiply each term. In our previous geometric sequence example, 2, 6, 18, 54, the common ratio is 3.

• Divide the second term by the first: \( \frac{6}{2} = 3 \)
• Apply the same process to other pairs: \( \frac{18}{6} = 3 \)

The symbol \(r\) represents the common ratio, which plays a key role in determining how rapidly the sequence grows or shrinks. A common ratio greater than 1 results in a rapid rise of the sequence, known as exponential growth, whereas a ratio between 0 and 1 will cause it to decrease exponentially, leading to convergence towards zero.