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

Jul 24, 2016

$50$

#### Explanation:

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

Suppose the middle number was n

$n + \left(n - 1\right) + \left(n + 1\right) = 3 n$ for the sum of the 3 middle numbers

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

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

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$\textcolor{b r o w n}{\text{The middle number is the mean value.}}$

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

$\implies n = \frac{645}{15} = 43$

$\textcolor{b l u e}{\text{The middle number is 43}}$

The last $\underline{\text{number count}}$ from the middle will be:

$\frac{15 - 1}{2} = \frac{14}{2} = 7$

$\textcolor{b l u e}{\text{So the largest number is } 43 + 7 = 50}$
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Check:
First number is $43 - 7 = 36$

$\implies \text{ sum "=(36+50)/2xx15 = 645 larr" Check is correct}$

$\text{ } \uparrow$
$\text{ Mean value}$