Question

A sequence, a, ax, ax \(^{2}, \ldots \ldots \ldots\) ax \(^{n}\), has odd number of terms. Find its median. (1) \(\mathrm{ax}^{\mathrm{n}-1}\) (2) \(\mathrm{ax}^{\frac{\mathrm{n}}{2}-1}\) (3) \(a x^{\frac{n}{2}}\) (4) \(a x^{\frac{n}{2}+1}\)

Short answer

Answer: \(a x^{\frac{n}{2}}\)

Step-by-step solution

Identify the nth term of the sequence

The given sequence is \(a, ax, ax^2, \ldots, ax^n\). We can see that the nth term of the sequence can be written as: \[T_n = ax^n \] where n = 0, 1, 2, ..., n.

Find the number of terms in the sequence

Since the sequence has odd number of terms, we can represent the number of terms as \(2k+1\) where k is an integer. The terms can have indices starting from 0 to n, which means there are \(n+1\) terms in the sequence. Thus, we have: \[2k+1 = n+1\]

Find the median of the sequence

For a sequence with odd number of terms, the median is the middle term. To find the middle term, we can take the average of the indices of the first and last terms since the sequence is equally spaced. Note that indexing starts from zero. Then, the index of the first term is 0 and the index of the last term is n. The index of the middle term is: \[m = \frac{0+n}{2} = \frac{n}{2}\]

Substitute the middle term index in the general term formula

Now, we can find the median by using the index of the middle term and substituting it in the general term formula for the sequence. \[T_m = ax^{\frac{n}{2}}\]

Compare the result with the available options

Our result shows that the median of the given sequence is \(ax^{\frac{n}{2}}\). Comparing this with the given options, it matches with option 3. Hence the correct answer is: \(a x^{\frac{n}{2}}\).

Key concepts

Geometric Sequence

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. In simpler terms, every term in a geometric sequence can be calculated by multiplying the preceding term by the same constant.
For example, in the sequence 2, 4, 8, 16, ..., the common ratio is 2. This means each term in the sequence is two times the previous term. This pattern continues indefinitely as long as the common ratio remains consistent. The general formula for the nth term in a geometric sequence is given by:

• \(a_n = a \cdot r^{(n-1)}\)

where \(a\) is the first term, \(r\) is the common ratio, and \(n\) is the term number.

Mathematical Sequence

A mathematical sequence is an ordered list of numbers that may follow a certain pattern or rule. Sequences can be finite or infinite, depending on whether they have a defined start and end.
Common types of sequences include arithmetic sequences, where the difference between consecutive terms is constant, and geometric sequences, where each term is a constant multiple of the previous term.
Each type of sequence is defined by its own specific rule or formula, which allows for the prediction of future terms within the sequence. For instance, in the case of a geometric sequence, the general rule would involve multiplying by the common ratio, as previously explained.

Odd Number of Terms

In mathematical sequences, having an odd number of terms has specific implications, particularly when it comes to finding the median. An odd number is any integer not divisible by 2. Sequences described as having an odd number of terms are frequently denoted by an expression like \(2k+1\), where \(k\) is an integer.
This ensures that there is a central term in the sequence which can easily be identified as the median. For example, if the sequence has 5 terms, the median is the 3rd term because it sits right in the middle of the sequence. Having an odd number of terms simplifies the process of finding the median, as there's exactly one middle term.

Median Finding Process

The median in a sequence is the middle value when the terms are arranged in order. Finding the median involves identifying this central term. If a sequence has an odd number of terms, the median is the term situated exactly in the middle.
To do this with an odd number term sequence, you take the index positions of the first and last term and calculate their average. This gives you the position of the median term. For example, if the sequence runs from index 0 to \(n\), and the sequence has an odd number of terms, then the median term will have the index:

• \[m = \frac{0 + n}{2} = \frac{n}{2}\]

Substituting this index into the sequence's general formula yields the median. In geometric sequences, this ensures the calculated median term reflects the characteristic ratio-based pattern.