Question

If 645 is written as the sum of fifteen consecutive integers, what is the largest of the addends?

Answer 1

`50`

Explanation:

Lets investigate the number behaviour as we move outwards from the middle number

Suppose the middle number was n

`n+(n-1)+(n+1)=3n` for the sum of the 3 middle numbers

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Then the next sum as we move outwards 1 step would be:

`3n+(n-2)+(n+2) = 5n` for the sum of the 5 middle numbers
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Then the next sum as we move outwards 1 step would be:
`5n+(n-3)+(n+3)=7n` for the sum of the 7 middle numbers

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`color(brown)("The middle number is the mean value.")`

Consequently, as there are 15 numbers we have `15n=645`

`=> n=645/15 = 43`

`color(blue)("The middle number is 43")`

The last `ul("number count")` from the middle will be:

`(15-1)/2 = 14/2=7`

`color(blue)("So the largest number is "43+7=50)`
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Check:
First number is `43-7= 36`

`=>" sum "=(36+50)/2xx15 = 645 larr" Check is correct"`

`" "uarr`
`" Mean value"`