Question

Determine whether the sequence is arithmetic, geometric, or neither. \(13.5,40.5,121.5,364.5, \ldots\)

Short answer

The given sequence \(13.5, 40.5, 121.5, 364.5, \ldots\) is geometric.

Step-by-step solution

Determine if the Sequence is Arithmetic

For a sequence to be arithmetic, the difference between successive terms must be constant. So, calculate the differences between consecutive terms to see if they are the same. Difference between second and first term is \(40.5 - 13.5 = 27\). Difference between third and second term is \(121.5 - 40.5 = 81\). From this, you can see that the differences are not constant, therefore, the sequence is not arithmetic.

Determine if the Sequence is Geometric

To identify if a sequence is geometric, the ratio of successive terms should be constant. Calculate the ratios between consecutive terms to check this. The ratio of the second term to the first is \(40.5 / 13.5 = 3\). Similarly, the ratio of the third term to the second is \(121.5 / 40.5 = 3\). As the ratios are the same, the sequence is geometric.

Key concepts

Understanding Arithmetic Sequences

An arithmetic sequence is a sequence of numbers in which each term after the first is found by adding a fixed, constant number called the "common difference" to the previous term. In simpler words, if you keep adding the same number to each term to get the next one, you have an arithmetic sequence. For example:

• An arithmetic sequence might start at 2 and have a common difference of 3, giving you 2, 5, 8, 11, and so on.

To test if a sequence is arithmetic, calculate the difference between each consecutive pair of terms:

• If the difference is the same all the way through, the sequence is arithmetic.
• If not, it's something else, like geometric or neither.

In our original problem, the sequence was tested by calculating the differences between the numbers in the sequence: 40.5-13.5=27 and 121.5-40.5=81, which aren't equal. Thus, it confirms the sequence is not arithmetic.

Decoding the Common Ratio in Sequences

The common ratio is a key identifier for geometric sequences. In a geometric sequence, each term is found by multiplying the previous term by a fixed, non-zero constant known as the "common ratio".

• For instance, in the sequence 3, 9, 27, 81, the common ratio is 3, because each term (after the first) is 3 times the previous term.

To determine if a sequence is geometric, calculate the ratio of each pair of consecutive terms:

• If all ratios are the same, you have found the common ratio and confirmed the sequence is geometric.
• If they vary, the sequence isn't geometric.

In the provided exercise, the sequence was checked for a constant ratio: 40.5/13.5=3 and 121.5/40.5=3. Since these ratios are equal, the sequence stands confirmed as geometric with a common ratio of 3.

Exploring Mathematical Sequences

Mathematical sequences are ordered lists of numbers, each of which is called a term of the sequence. They follow a certain pattern or rule, which defines three main types:

• Arithmetic sequences, defined by addition of a constant value.
• Geometric sequences, where each term is a constant multiple of the previous term.
• Other sequences, like the Fibonacci sequence, where the pattern is more complex, involving operations like sums of previous terms.

Understanding the fundamental types of sequences helps to solve various mathematical problems by identifying patterns or rules based on given terms. Recognizing the type of sequence can help predict future terms and understand the inherent pattern. In our example, determining the sequence is geometric assisted us in predicting subsequent terms by simply multiplying the common ratio by the latest term. These foundational concepts provide a stepping stone to understanding more complex mathematical ideas.