Question 141060
The 15th term of an arithmetic series is 52, and the sum of the first 15 terms is 405. Find the first term of the series. 
<pre><font size = 4 color = "indigo"><b>
Your book may use the letter "a" where I use the letter "t".
Some books use "t" and some "a".

Given: n=15, t<sub>15</sub> = 52,  S<sub>15</sub> = 405

You need two formulas.

1. The nth term of an arithmetic series is given by the formula

   t<sub>n</sub> = t<sub>1</sub> + (n-1)d

2. The sum of the first n terms of an arithmetic series is given by
   the formula

   S<sub>n</sub> = {{{n/2}}}[2t<sub>1</sub> + (n-1)d]

Substituting in the first formula:

   t<sub>15</sub> = t<sub>1</sub> + (15-1)d
     52 = t<sub>1</sub> + 14d


Substituting in the second formula:

   S<sub>15</sub> = {{{15/2}}}[2t<sub>1</sub> + (15-1)d]

    405 = {{{15/2}}}(2t<sub>1</sub> + 14d)

Clear of fractions by multiplying both sides by 2

    810 = 15(2t<sub>1</sub> + 14d)

Divide both sides by 15

     54 = 2t<sub>1</sub> + 14d

So you have this system of equations:


     52 = t<sub>1</sub> + 14d
     54 = 2t<sub>1</sub> + 14d

Can you solve that system using substitution
or elimination?  If not post again asking how.

Answer:       t<sub>1</sub> = 2. d = 25/7  

Edwin</pre>