Question

The median of a set of 55 consecutive odd integers is 55. What is the greatest of these integers?

Answer 1

Hint : sum of first \(n\) odd positive integers equals the perfect square, \(n^2\).

Answer 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 :
\[\{1,3,5,\ldots, 2n-1\}\]

plugin \(n=55\) to get the greatest integer :
\(2n-1=2*55-1 =109 \)

Answer 3

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
etc


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

Answer 4

How do we know upfront that the smallest element in the set is 1 ?

Answer 5

I'm using the info `The median of a set of 55 consecutive odd integers is 55`

Answer 6

That doesn't tell us that the smallest integer in the set is 1
and it seems the previous method is assuming that the smallest integer is 1, right ?

Answer 7

I never said that the smallest integer in that set is 1

Answer 8

but I just generated a list and the smallest integer is 1

Answer 9

Ahh nice, got you ! my mistake, sorry...