A sequence a, ax, ax 2, _______ ax n, has odd number of terms. Then the median is

Question

Prepp Team · Accepted Answer

Understanding the Sequence and the Problem

The question asks for the median of a sequence given as $ a, ax, ax^2, \dots, ax^n $. We are told that this sequence has an odd number of terms.

Let's first identify the terms in the sequence and the total number of terms.

The first term is $ a $, which can be written as $ ax^0 $.
The second term is $ ax $, which can be written as $ ax^1 $.
The third term is $ ax^2 $.
...
The last term is $ ax^n $.

The powers of $ x $ in the terms range from 0 to $ n $. The terms are indexed by the power of $ x $ plus one. For example, the term with $ x^k $ is the $(k+1)$-th term.

The total number of terms in this sequence is the number of possible values for the power of $ x $, which are $ 0, 1, 2, \dots, n $. This gives a total of $ n - 0 + 1 = n + 1 $ terms.

Handling the Odd Number of Terms

We are given that the sequence has an odd number of terms. The total number of terms is $ n+1 $. If $ n+1 $ is an odd number, it means that $ n $ must be an even number. For instance, if $ n=0 $, terms = 1 (odd); if $ n=2 $, terms = 3 (odd); if $ n=4 $, terms = 5 (odd). So, $ n $ must be even.

Let $ N $ be the total number of terms. Here, $ N = n+1 $.

For a sequence with an odd number of terms, the median is the middle term when the terms are arranged in ascending or descending order. Since this is a geometric sequence with common ratio $ x $, the order depends on whether $ x > 1 $ (ascending) or $ 0 < x < 1 $ (descending, assuming $ a > 0 $). However, regardless of the order, the middle term remains the same element from the set of terms.

Finding the Position of the Median Term

For a sequence with $ N $ terms where $ N $ is odd, the position of the median term is given by the formula:

$ \text{Median Position} = \frac{N+1}{2} $

In our case, the number of terms is $ N = n+1 $. So, the position of the median term is:

$ \text{Median Position} = \frac{(n+1)+1}{2} = \frac{n+2}{2} $

Determining the Median Term

Now we need to find the value of the term located at the position $ \frac{n+2}{2} $.

Let's look at the terms and their positions again:

1st term: $ ax^0 $
2nd term: $ ax^1 $
3rd term: $ ax^2 $
...
$ p $-th term: $ ax^{p-1} $

The general formula for the $ p $-th term is $ ax^{p-1} $. We found that the median is at position $ p = \frac{n+2}{2} $. Let's substitute this position into the formula for the $ p $-th term:

$ \text{Median Term} = ax^{\left(\frac{n+2}{2}\right) - 1} $

Now, let's simplify the exponent:

$ \left(\frac{n+2}{2}\right) - 1 = \frac{n+2}{2} - \frac{2}{2} = \frac{n+2-2}{2} = \frac{n}{2} $

So, the median term is $ ax^{\frac{n}{2}} $.

Since $ n $ is an even number (as $ n+1 $ is odd), $ \frac{n}{2} $ is an integer, which makes sense as the power of $ x $ in the terms are integers from 0 to $ n $.

Comparing with Options

Let's compare our result $ ax^{\frac{n}{2}} $ with the given options:

$ {\rm{a}}{{\rm{x}}^{\frac{{\rm{n}}}{2} + 1}} $
$ {\rm{a}}{{\rm{x}}^{\frac{{\rm{n}}}{2} - 1}} $
$ {\rm{a}}{{\rm{x}}^{{\rm{n}} - 1}} $
$ {\rm{a}}{{\rm{x}}^{\frac{{\rm{n}}}{2}}} $

Our calculated median term matches option 4.

Concept	Description	Formula/Example
Sequence	An ordered list of numbers.	$ a, a_2, a_3, \dots $
Geometric Sequence	A sequence where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio ($ x $).	$ a, ax, ax^2, ax^3, \dots $
Number of Terms	The count of elements in the sequence.	$ n+1 $ for the sequence $ a, ax, \dots, ax^n $
Median	The middle value in a dataset that is ordered from least to greatest.	For odd number of terms $ N $, it's the term at position $ \frac{N+1}{2} $.

Revision Table: Key Steps to Find the Median

Step	Action	Explanation
1	Identify the sequence and number of terms.	Sequence is $ a, ax, \dots, ax^n $. Number of terms is $ n+1 $.
2	Note that the number of terms is odd.	This means $ n+1 $ is odd, which implies $ n $ is even.
3	Determine the median position.	For $ N $ terms (odd), position is $ \frac{N+1}{2} $. Substitute $ N=n+1 $.
4	Calculate the median position.	Position $ = \frac{(n+1)+1}{2} = \frac{n+2}{2} $.
5	Find the term at the median position.	The $ p $-th term of the sequence $ a, ax, \dots, ax^n $ is $ ax^{p-1} $.
6	Substitute the median position into the term formula.	Term $ = ax^{\left(\frac{n+2}{2}\right) - 1} = ax^{\frac{n}{2}} $.

Additional Information: Median vs. Mean

It's important not to confuse the median with the mean (average) of a sequence. The mean is calculated by summing all terms and dividing by the number of terms. The median, on the other hand, is a positional average; it is simply the value of the middle term when the sequence is ordered.

For a sequence with an even number of terms, the median is typically defined as the average of the two middle terms. However, in this problem, we are specifically told that the number of terms is odd.

The terms in the sequence $ a, ax, ax^2, \dots, ax^n $ are already in order (assuming $ x > 0 $; if $ x < 0 $, the order might alternate, but the set of values and thus the middle value remains the same). The power of $ x $ increases steadily from 0 to $ n $. The median term is the one with the middle power of $ x $.

Since the powers are $ 0, 1, 2, \dots, n $, and $ n $ is even, the middle power is exactly $ \frac{n}{2} $. The term with this power is $ ax^{n/2} $.

Prepp Team · Answer

${\rm{a}}{{\rm{x}}^{\frac{{\rm{n}}}{2} + 1}}$

Prepp Team · Answer

${\rm{a}}{{\rm{x}}^{\frac{{\rm{n}}}{2} - 1}}$

Prepp Team · Answer

${\rm{a}}{{\rm{x}}^{{\rm{n}} - 1}}$

Understanding the Sequence and the Problem

Handling the Odd Number of Terms

Finding the Position of the Median Term

Determining the Median Term

Comparing with Options

Revision Table: Key Steps to Find the Median

Additional Information: Median vs. Mean

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Concept	Description	Formula/Example
Sequence	An ordered list of numbers.	\( a, a_2, a_3, \dots \)
Geometric Sequence	A sequence where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio (\( x \)).	\( a, ax, ax^2, ax^3, \dots \)
Number of Terms	The count of elements in the sequence.	\( n+1 \) for the sequence \( a, ax, \dots, ax^n \)
Median	The middle value in a dataset that is ordered from least to greatest.	For odd number of terms \( N \), it's the term at position \( \frac{N+1}{2} \).

Step	Action	Explanation
1	Identify the sequence and number of terms.	Sequence is \( a, ax, \dots, ax^n \). Number of terms is \( n+1 \).
2	Note that the number of terms is odd.	This means \( n+1 \) is odd, which implies \( n \) is even.
3	Determine the median position.	For \( N \) terms (odd), position is \( \frac{N+1}{2} \). Substitute \( N=n+1 \).
4	Calculate the median position.	Position \( = \frac{(n+1)+1}{2} = \frac{n+2}{2} \).
5	Find the term at the median position.	The \( p \)-th term of the sequence \( a, ax, \dots, ax^n \) is \( ax^{p-1} \).
6	Substitute the median position into the term formula.	Term \( = ax^{\left(\frac{n+2}{2}\right) - 1} = ax^{\frac{n}{2}} \).