# I don't understand what my math teacher is teaching... How am I suppose to find "n" if I am given a1=-7 d=1.5 and Sn=-14 or a1=-3 d=2.5 and Sn=31.5? I tried everything but I just don't get it

steveschoen | Certified Educator

Hi, Shae,

I believe I can assist.  It may not be your exact problem.  But, I hope it can assist.

There are two ways to do this.  One way looks like to me is to solve a system of equations.  The other way would be to write the series and add each consecutive number.  As in:

a1 = -7

a1+d = a2 = -5.5     Sn = -7 + -5.5 = -12.5

a2+d = a3 = -4        Sn = -7 + -5.5 + -4 = -16.5

a3+d = a4 = -2.5     Sn = -7 + -5.5 + -4 + -2.5 = -19

a4+d = a5 = -1        Sn = -7 + -5.5 + -4 + -2.5 + -1 = -20

a5+d = a6 = 0.5       Sn = -7 + -5.5 + -4 + -2.5 + -1 + 0.5 = -19.5

a6+d = a7 = 2          Sn = -7 + -5.5 + -4 + -2.5 + -1 + 0.5 + 2 = -17.5

a7+d = a8 = 3.5       Sn = -7 + -5.5 + -4 + -2.5 + -1 + 0.5 + 2 + 3.5 = -14

So, we needed 8 terms (n = 8) for the sum, Sn, to be -14.  So:

n = 8

For the second one, doing the exact same method, we would go to a7 = a6 + d = 9.5 + 2.5 = 12.  Because, then, adding for the second one:

a1 + a2 + a3 + a4 + a5 + a6 + a7 = 31.5

-3 + -0.5 + 2 + 4.5 + 7 + 9.5 + 12 = 31.5

So, n has to be 7.

Good luck, Shae.  I hope this helps.

Till Then,

Steve