If #a, b, c, d# are in harmonic progress then which of the following is equal to #ab+bc+cd# ?

A. #3ad#
B. #3bc#
C. #3bd#
D. #3ac#

1 Answer
George C.
Apr 30, 2016

A. #3ad#

Explanation:

If #a, b, c, d# are in harmonic progression then their reciprocals are in arithmetic progression and we can write:

#{(a = 1/A), (b = 1/(A+D)), (c = 1/(A+2D)), (d = 1/(A+3D)) :}#

for some constants #A# and #D#.

Then we find:

#ab+bc+cd#

#=1/(A(A+D))+1/((A+D)(A+2D))+1/((A+2D)(A+3D))#

#=((A+2D)(A+3D)+A(A+3D)+A(A+D))/(A(A+D)(A+2D)(A+3D))#

#=(3A^2+9AD+6D^2)/(A(A+D)(A+2D)(A+3D))#

#=(3(A^2+3AD+2D^2))/(A(A+D)(A+2D)(A+3D))#

#=(3color(red)(cancel(color(black)((A+D))))color(red)(cancel(color(black)((A+2D)))))/(Acolor(red)(cancel(color(black)((A+D))))color(red)(cancel(color(black)((A+2D))))(A+3D))#

#=3/(A(A+3D))#

#=3*1/A*1/(A+3D)#

#=3ad#

Related questions
Impact of this question
541 views around the world
You can reuse this answer
Creative Commons License