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

Question

Kennedy Adams · Accepted Answer

#50#

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

'.............................................................................................. 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 '................................................................................................. 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

,~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ #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)# '~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Check: First number is #43-7= 36#

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

#" "uarr# # " Mean value"#

Genesis Allen · Answer

undefined The largest of the addends is 679.