<PRE><B>Question 830902</B>
&lt;pre&gt;
a&lt;sub&gt;1&lt;/sub&gt;, a&lt;sub&gt;2&lt;/sub&gt;, a&lt;sub&gt;3&lt;/sub&gt;, a&lt;sub&gt;4&lt;/sub&gt;, a&lt;sub&gt;5&lt;/sub&gt;, a&lt;sub&gt;6&lt;/sub&gt;, a&lt;sub&gt;7&lt;/sub&gt;, a&lt;sub&gt;8&lt;/sub&gt;, a&lt;sub&gt;9&lt;/sub&gt;, a&lt;sub&gt;10&lt;/sub&gt;, a&lt;sub&gt;11&lt;/sub&gt;

You need this formula for the nth term:

a&lt;sub&gt;n&lt;/sub&gt; = a&lt;sub&gt;1&lt;/sub&gt; + (n-1)d

[Some books and teachers use &quot;t&quot; for &quot;term&quot; instead of &quot;a&quot;.
I&#39;ll use &quot;a&quot;.]

&lt;/pre&gt;
The 11th term of an arithmetic sequence is 57 
&lt;pre&gt;
a&lt;sub&gt;11&lt;/sub&gt; = a&lt;sub&gt;1&lt;/sub&gt; + (11-1)d
57 = a&lt;sub&gt;1&lt;/sub&gt; + 10d
&lt;/pre&gt;
and the sum of the first and the fourth terms is 29.
&lt;pre&gt;
a&lt;sub&gt;1&lt;/sub&gt; + a&lt;sub&gt;4&lt;/sub&gt; = 29

Substitute n=4 in the formula also

a&lt;sub&gt;4&lt;/sub&gt; = a&lt;sub&gt;1&lt;/sub&gt; + (4-1)d
a&lt;sub&gt;4&lt;/sub&gt; = a&lt;sub&gt;1&lt;/sub&gt; + 3d

Now substitute for a&lt;sub&gt;4&lt;/sub&gt; in 

a&lt;sub&gt;1&lt;/sub&gt; + a&lt;sub&gt;4&lt;/sub&gt; = 29

a&lt;sub&gt;1&lt;/sub&gt; + (a&lt;sub&gt;1&lt;/sub&gt; + 3d) = 29 

a&lt;sub&gt;1&lt;/sub&gt; + a&lt;sub&gt;1&lt;/sub&gt; + 3d = 29

2a&lt;sub&gt;1&lt;/sub&gt; + 3d = 29

So you have this system of equations:

57 = a&lt;sub&gt;1&lt;/sub&gt; + 10d
2a&lt;sub&gt;1&lt;/sub&gt; + 3d = 29

Solve the first for a&lt;sub&gt;1&lt;/sub&gt;

57-10d = a&lt;sub&gt;1&lt;/sub&gt;

Substitute in 

2a&lt;sub&gt;1&lt;/sub&gt; + 3d = 29
2(57-10d) + 3d = 29
114 - 20d + 3d = 29
     114 - 17d = 29
          -17d = -85
             d = {{{(-85)/(-17)}}}
             d = 5

Substitute d=5 in 

57-10d = a&lt;sub&gt;1&lt;/sub&gt;
57-10(5) = a&lt;sub&gt;1&lt;/sub&gt;
57-50 = a&lt;sub&gt;1&lt;/sub&gt;
7 = a&lt;sub&gt;1&lt;/sub&gt;

So the sequence starts with a&lt;sub&gt;1&lt;/sub&gt;=7 and adds d=5 each time.

To check, let&#39;s write them all out and see if the llth terms comes out
to be 57. 

7, 12, 17, 22, 27, 32, 37, 42, 47, 52, 57

The last one came out 57.  Count them.  There are 11.  So it&#39;s correct. 
&lt;/pre&gt;
Determine the first 3 terms of the sequence.
&lt;pre&gt;
7, 12, 17

Edwin&lt;/pre&gt; </PRE>