Question

Write the next two apparent terms of the sequence. Describe the pattern you used to find these terms. \(1,-\frac{3}{2}, \frac{9}{4},-\frac{27}{8}, \ldots\)

Short answer

Therefore, the next two terms in the sequence are \(81/16\) and \(-243/32\). The pattern used to find these terms involves an alternating sign and the absolute value of the terms being successive powers of 3 divided by successive powers of 2.

Step-by-step solution

Identify the Pattern in the Signs

Look at the signs of the given sequence. They alternate between positive and negative. Thus, the next two terms should also follow this alternating pattern. The last term in the sequence is negative so the next term should be positive.

Identify the Pattern in the Numbers

Apart from the signs, look at the absolute value of the numbers in the sequence: 1, 3/2, 9/4, and 27/8. This can be seen as 3^0/2^0, 3^1/2^1, 3^2/2^2, and 3^3/2^3. The numerator and denominator in the fractions seem to be increasing by powers of 3 and 2 respectively. Thus, the next two terms should follow that pattern as well. The fifth term in the sequence should be \(3^4/2^4\) and the sixth term in the sequence should be \(3^5/2^5\)

Combine the Patterns

Merge the two identified patterns. The fifth term (from Step 1 and Step 2) is positive, and the absolute value of the number is \(3^4/2^4\), which simplifies to \(81/16\). The sixth term is negative, and the absolute value of the number is \(3^5/2^5\), which simplifies to \(243/32\)

Key concepts

Sequence Patterns

Understanding sequence patterns is crucial in mathematics as they form the basis for series and other complex functions. A sequence is an ordered list of numbers; the pattern refers to the rule used to formulate each term in the sequence from the previous one, or from its position in the sequence.

For our exercise, the sequence starts with 1 and continues with -3/2, 9/4, -27/8, and so on. To determine the next terms in such a sequence, it's important to look for regularities both in the signs of the numbers (positive or negative) and their magnitudes (the numbers themselves).

Identifying Sequence Patterns

In the given sequence, two distinct patterns emerge. The signs alternate between positive and negative, creating an alternating series, while the magnitudes of the terms follow an exponential pattern. Recognizing these patterns allows us to predict future terms. In short, understanding and identifying sequence patterns forms a fundamental skill in problem-solving within mathematics.

Alternating Series

An alternating series is a particular type of sequence where the signs of the terms switch between positive and negative from one term to the next. This property is used in mathematical analysis to describe a series whose terms alternate in sign, often resulting in interesting convergence properties.

The key to understanding an alternating series is to focus on the change of sign between consecutive terms. In our exercise, terms alternate signs starting with a positive 1, then -3/2, and so on. The next term after -27/8 must be positive because the pattern is to alternate.

Application in Sequences

The concept of an alternating series is essential when we look at infinite series, particularly when assessing their convergence or divergence. Alternating series can also be used in calculus to approximate functions and in quantitative finance to calculate the present value of alternating cash flows.

Exponential Expressions

Exponential expressions are used to describe patterns where each term involves a number raised to some power – usually the term's position in the sequence. These expressions can model growth or decay processes and are integral to fields ranging from finance to physics.

Looking at the absolute values in our exercise: 1, 3/2, 9/4, 27/8,... we can express them using exponents as 3^0/2^0, 3^1/2^1, 3^2/2^2, 3^3/2^3, etc. The pattern here is that with each successive term, the numerator and denominator are raised to progressively higher powers of 3 and 2, respectively.

Recognizing Exponential Growth

To predict future terms, understanding how to work with exponential expressions is vital. It's about recognizing the base and the way the exponent increases. In our case, the base for the numerator is 3 and for the denominator is 2. The exponents match the term's order in the sequence and are used to calculate the next terms, ensuring accuracy in the sequence's continuation.