The second term of an arithmetic sequence is 24 and the fifth term is 3. What is the first term and the common difference?

1 Answer
Nov 8, 2017

First term #31# and common difference #-7#

Explanation:

Let me start by saying how you might really do this, then showing you how you should do it...

In going from the 2nd to the 5th term of an arithmetic sequence, we add the common difference #3# times.

In our example that results in going from #24# to #3#, a change of #-21#.

So three times the common difference is #-21# and the common difference is #-21/3 = -7#

To get from the 2nd term back to the 1st one, we need to subtract the common difference.

So the first term is #24-(-7) = 31#

So that was how you might reason it. Next let's see how to do it a little more formally...

The general term of an arithmetic sequence is given by the formula:

#a_n = a+d(n-1)#

where #a# is the initial term and #d# the common difference.

In our example we are given:

#{ (a_2 = 24), (a_5 = 3) :}#

So we find:

#3d = (a+4d) - (a+d)#

#color(white)(3d) = (a+(5-1)d) - (a+(2-1)d)#

#color(white)(3d) = a_5 - a_2#

#color(white)(3d) = 3-24#

#color(white)(3d) = -21#

Dividing both ends by #3# we find:

#d = -7#

Then:

#a = a_1 = a_2-d = 24-(-7) = 31#