<PRE><B>Question 412077</B>
Hello, can somebody please help me with this question I don&#39;t understand it

The 8th term of an arithmetic sequence is 25 and the 20th term is -11. Find the first term of the sequence and the common difference and use those to write the general term.

&lt;pre&gt;&lt;font face = &quot;consolas&quot; color = &quot;indigo&quot; size = 4&gt;&lt;b&gt;

The other tutor&#39;s answer is wrong.


__, __, __, __, __, __, __, 25, __, __, __, __, __, __, __, __, __, __, __, -11, __, ...   

The general term of any arithmetic sequence is the formula
for the nth term which is

a&lt;sub&gt;n&lt;/sub&gt; = a&lt;sub&gt;1&lt;/sub&gt; + (n - 1)d

Substitute 8 for n:

a&lt;sub&gt;8&lt;/sub&gt; = a&lt;sub&gt;1&lt;/sub&gt; + (8 - 1)d

a&lt;sub&gt;8&lt;/sub&gt; = a&lt;sub&gt;1&lt;/sub&gt; + 7d

Substitute 25 for a&lt;sub&gt;8&lt;/sub&gt;

25 = a&lt;sub&gt;1&lt;/sub&gt; + 7d

a&lt;sub&gt;1&lt;/sub&gt; + 7d = 25

----------------

a&lt;sub&gt;n&lt;/sub&gt; = a&lt;sub&gt;1&lt;/sub&gt; + (n - 1)d

Substitute 20 for n:

a&lt;sub&gt;20&lt;/sub&gt; = a&lt;sub&gt;1&lt;/sub&gt; + (20 - 1)d

a&lt;sub&gt;20&lt;/sub&gt; = a&lt;sub&gt;1&lt;/sub&gt; + 19d

Substitute -11 for a&lt;sub&gt;20&lt;/sub&gt;

-11 = a&lt;sub&gt;1&lt;/sub&gt; + 19d

a&lt;sub&gt;1&lt;/sub&gt; + 19d = -11

Now we have the system of equations:

a&lt;sub&gt;1&lt;/sub&gt; +  7d =  25
a&lt;sub&gt;1&lt;/sub&gt; + 19d = -11

Solve the first one for a&lt;sub&gt;1&lt;/sub&gt;
      a&lt;sub&gt;1&lt;/sub&gt; =  25 - 7d

Substitute (25 - 7d) for a&lt;sub&gt;1&lt;/sub&gt;

(25 - 7d) + 19d = -11
  25 - 7d + 19d = -11
       25 + 12d = -11
            12d = -36
              d = -3

             a&lt;sub&gt;1&lt;/sub&gt; =  25 - 7d
             a&lt;sub&gt;1&lt;/sub&gt; =  25 - 7(-3)
             a&lt;sub&gt;1&lt;/sub&gt; =  25 + 21
             a&lt;sub&gt;1&lt;/sub&gt; =  46

So a&lt;sub&gt;1&lt;/sub&gt; = 46 is the first term of the sequence and d = -3 is the common
difference.   

So the arithmetic sequence is:

46, 43, 40, 37, 34, 31, 28, 25, 22, 19, 16, 13, 10, 7, 4, 1, -2, -5, -8, -11, -14, ...

The general term is found by substituting a&lt;sub&gt;1&lt;/sub&gt; = 46 and d = -3 in the
formula for the nth term:

a&lt;sub&gt;n&lt;/sub&gt; = a&lt;sub&gt;1&lt;/sub&gt; + (n - 1)d

a&lt;sub&gt;n&lt;/sub&gt; = 46 + (n - 1)(-3)

a&lt;sub&gt;n&lt;/sub&gt; = 46 - 3(n - 1)

a&lt;sub&gt;n&lt;/sub&gt; = 46 - 3n + 3

a&lt;sub&gt;n&lt;/sub&gt; = 49 - 3n

That&#39;s the general term.

Edwin&lt;/pre&gt; </PRE>