anonymous
  • anonymous
The median of a set of 55 consecutive odd integers is 55. What is the greatest of these integers?
Mathematics
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SOLVED
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chestercat
  • chestercat
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ganeshie8
  • ganeshie8
Hint : sum of first \(n\) odd positive integers equals the perfect square, \(n^2\). |dw:1446852815273:dw|
ganeshie8
  • ganeshie8
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 : \(2n-1=2*55-1 =109 \)
jim_thompson5910
  • jim_thompson5910
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

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ganeshie8
  • ganeshie8
How do we know upfront that the smallest element in the set is 1 ?
jim_thompson5910
  • jim_thompson5910
I'm using the info `The median of a set of 55 consecutive odd integers is 55`
ganeshie8
  • ganeshie8
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 ?
jim_thompson5910
  • jim_thompson5910
I never said that the smallest integer in that set is 1
jim_thompson5910
  • jim_thompson5910
but I just generated a list and the smallest integer is 1
ganeshie8
  • ganeshie8
Ahh nice, got you ! my mistake, sorry...

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