SOLUTION: The xth, yth, and zth terms of a sequence are X,Y,Z respectively. Show that if the sequence is arithmetic then X(y-z) + Y(z-x) + Z(x-y)=0.

Algebra ->  Sequences-and-series -> SOLUTION: The xth, yth, and zth terms of a sequence are X,Y,Z respectively. Show that if the sequence is arithmetic then X(y-z) + Y(z-x) + Z(x-y)=0.       Log On


   



Question 1110008: The xth, yth, and zth terms of a sequence are X,Y,Z respectively. Show that if the sequence is arithmetic then X(y-z) + Y(z-x) + Z(x-y)=0.

Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
Let it be that in an arithmetic sequence
a= first term of the arithmetic sequence
d= common difference of the arithmetic sequence.
Then, if the xth, yth, and zth terms of athat sequence are X,Y,Z respectively,
X=a%2B%28x-1%29d=a%2Bxd-d ,
Y=a%2B%28y-1%29d=a%2Byd-d , and
Z=a%2B%28z-1%29d=a%2Bzd-d .
X%28y-z%29+%2B+Y%28z-x%29+%2B+Z%28x-y%29 %22=%22 %28a%2Bxd-d%29%28y-z%29%2B%28a%2Byd-d%29%28z-x%29%2B%28a%2Bzd-d%29%28x-y%29
%22=%22 a%28y-z%2Bz-x%2Bx-y%29%22%2B%22%22%5B%22xd%28y-z%29%2Byd%28z-x%29%2Bzd%28x-y%29%22%5D%22%22-%22d%28y-z%2Bz-x%2Bx-y%29
%22=%22 a%2A0%2Bd%28xy-xz%2Byz-yx%2Bzx-zy%29-d%2A0 %22=%22 0%2Bd%2A0%2B0 %22=%22 0%2B0%2B0 %22=%22 0