If #x+14, 13-x, x+8# is an arithmetic sequence, then what is the value of #x# ?

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Answer 1 · 2017-05-19T21:14:31

#x=1#

The difference between #13x-1# and #x+14# is:

#(13x-1)-(x+14) = 12x-15#

The difference between #x+8# and #13x-1# is:

#(x+8) - (13x-1) = -12x+9#

The given terms form an arithmetic sequence if and only if these two differences are equal. That is:

#12x-15 = -12x+9#

Add #12x+15# to both sides to get:

#24x=24#

Divide both sides by #24# to get:

#x=1#

So this is the only solution, yielding the arithmetic sequence:

#15, 12, 9#

Answer 2 · 2017-05-21T15:00:32

#x=1#

As this gives a linear equation, there is only one solution.

If you know it is an arithmetic sequence, then you know that the common difference, #d# between any two consecutive terms is the same.

#d= T_3-T_2= T_2-T_1#

#d = (x+8)- (13x-1)= (13x-1)-(x+14)#

#color(white)(............)x+8-13x+1 = 13x-1-x-14#

#color(white)(............................)9+15 = 12x+12x#

#color(white)(..................................)24 = 24x#

#color(white)(.....................................)1=x#