Question

Find the median of the first 15 odd numbers.

Answer 1

Hint: In simple terms. Median is the middle term. So, write the first 15 odd numbers. Then find the middle term to get the median. You can use the formula for finding the middle term. Remember that the formula of finding the middle term is different for the odd number of terms and even number of terms.

Complete step-by-step answer:
In simple terms, we can say that the median is the middle term. For a small sequence, we can simply count the middle term and write the answer. But for larger sequences, we use the formula for the middle term in terms of number of terms of the sequence.
The formula is given by
 $ median = {\left( {\dfrac{{n + 1}}{2}} \right)^{th}} $ term if $ n $ is odd. And
 $ median = \dfrac{{{{\left( {\dfrac{n}{2}} \right)}^{th}}term + {{\left( {\dfrac{n}{2} + 1} \right)}^{th}}term}}{2} $ if $ n $ is even.
The term can be found by either counting it, or by using the formulae of $ {n^{th}} $ term of sequence.
Now, in the question, it is given that we have the sequence of the first 15 odd numbers. It can be written as
 $ S = \{ 1,3,5,7,...,15\} $
When we observe, we can say that every term of this sequence is represented by the formula,
 $ {t_n} = 2n - 1 $
where $ n $ is the number of term
since, there are 15 terms and 15 is an odd number. The formula we use to find the median will be
 $ median = {\left( {\dfrac{{n + 1}}{2}} \right)^{th}}term $
Now, $ {\left( {\dfrac{{n + 1}}{2}} \right)^{th}}term $ is $ {t_{\dfrac{{n + 1}}{2}}} $ . i.e.
 $ {t_{\dfrac{{n + 1}}{2}}} = 2\left( {\dfrac{{n + 1}}{2}} \right) - 1 $
Here $ n = 15 $
Thus the median will be
 $ {t_{\dfrac{{15 + 1}}{2}}} = 2\left( {\dfrac{{15 + 1}}{2}} \right) - 1 $
 $ \Rightarrow {t_8} = 16 - 1 $
 $ \Rightarrow median = {t_8} = 15 $
Therefore, the median of the first 15 odd numbers is 15.
So, the correct answer is “15”.

Note: You need to know the fact that the formula of median is different for even number of terms and odd number of terms. The key point for solving such a question is to write the given sequence in the form of a general term, in terms of $ n $ . Otherwise, counting the middle term will be very difficult. Especially, when the sequence is large. For example, what if the question was, find the median of the first 150 odd numbers?