The median of a set of 55 consecutive odd integers is 55. What is the greatest of these integers?
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Hint : sum of first \(n\) odd positive integers equals the perfect square, \(n^2\).
You're given that the median of \(55\) consecutive odd integers is \(55\). That means the sum of those 55 consecutive odd integers must equal \(55^2\). It follows, the set is simply the first \(55\) positive odd integers :
plugin \(n=55\) to get the greatest integer :
\(2n-1=2*55-1 =109 \)
Or you can think smaller
Say we had 3 items. The middle term is in slot 2
Say we had 5 items. The middle term is in slot 3
Say we had 7 items. The middle term is in slot 4
The midpoint slot is equal to (n+1)/2
For instance, if n = 5, then (n+1)/2 = (5+1)/2 = 3 is the slot number for the midpoint
Now n = 55, so (n+1)/2 = (55+1)/2 = 56/2 = 28
There are 55-28 = 27 more numbers after the midpoint
Each term is found by adding 2 each time
55+2 = 57
57+2 = 59
In general, the nth term after 55 is 55+2n
Plug in n = 27 to get
55+2*27 = 109
and you get the same answer