# If #a+c=2b# and #ab+cd+ad=3bc#, how do I show that #a,b,c,d# are in arithmetic progression?

##### 1 Answer

Play around with the equations. Knowing how to (re-)phrase your goal mathematically can help you deduce it from the given information.

#### Explanation:

**if and only if**

Can we show this?

Given:

Move one

#color(white)(=>)a+c=2b#

#=>a-b=b-c" "# Multiply both sides by#-1# .

#=>b-a=c-b#

Now we just need to show that

Given:

Solve for

#ab+cd+ad=3bc#

#=>d(c+a)=3bc-ab#

#=>d=(b(3c-a))/(c+a)" "# Substitute#2b=a+c#

#=>d=(cancel b(3c-a))/(2cancelb)#

#=>2d=3c-a" "# Subtract#2c# from both sides

#=>2d-2c=3c-a-2c#

#=>2(d-c)=c-a" "# Subtractandadd#b# to the RHS

#=>2(d-c)=c-b+b-a" "# Substitute#b-a=c-b#

#=>2(d-c)=c-b+c-b#

#=>2(d-c)=2(c-b)" "# Divide both sides by 2

#=>d-c=c-b#

Since