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
First term
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
In our example that results in going from
So three times the common difference is
To get from the 2nd term back to the 1st one, we need to subtract the common difference.
So the first term is
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
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
#d = -7#
Then:
#a = a_1 = a_2-d = 24-(-7) = 31#