<PRE><B>Question 142221</B>
find the explicit formula 
a&lt;sub&gt;7&lt;/sub&gt;=-10 and a&lt;sub&gt;13&lt;/sub&gt;=-48
&lt;pre&gt;&lt;font size = 4 color = &quot;indigo&quot;&gt;&lt;b&gt;
The problem is to find a formula that we can use to 
fill in the empty blanks of this sequence:

___,___,___,___,___,___,&lt;u&gt;-10&lt;/u&gt;,___,___,___,___,___,&lt;u&gt;-48&lt;/u&gt;


Let&#39;s assume it&#39;s an arithmetic sequence.  Then we use

a&lt;sub&gt;n&lt;/sub&gt; = a&lt;sub&gt;1&lt;/sub&gt; + (n-1)d

Substitute n=7

a&lt;sub&gt;n&lt;/sub&gt; = a&lt;sub&gt;1&lt;/sub&gt; + (7-1)d
a&lt;sub&gt;7&lt;/sub&gt; = a&lt;sub&gt;1&lt;/sub&gt; + 6d

when n=7, a&lt;sub&gt;n&lt;/sub&gt; = -10, so a&lt;sub&gt;7&lt;/sub&gt; = -10

Substituting a&lt;sub&gt;7&lt;/sub&gt; = -10

-10 = a&lt;sub&gt;1&lt;/sub&gt; + 6d

Also,

a&lt;sub&gt;n&lt;/sub&gt; = a&lt;sub&gt;1&lt;/sub&gt; + (n-1)d

Substitute n=13

a&lt;sub&gt;13&lt;/sub&gt; = a&lt;sub&gt;1&lt;/sub&gt; + (13-1)d
a&lt;sub&gt;13&lt;/sub&gt; = a&lt;sub&gt;1&lt;/sub&gt; + 12d

when n=13, a&lt;sub&gt;n&lt;/sub&gt; = -48, so a&lt;sub&gt;13&lt;/sub&gt; = -48

Substituting a&lt;sub&gt;13&lt;/sub&gt; = -48

-48 = a&lt;sub&gt;1&lt;/sub&gt; + 12d

So we have the system of two equations and
two unknowns

-10 = a&lt;sub&gt;1&lt;/sub&gt; + 6d
-48 = a&lt;sub&gt;1&lt;/sub&gt; + 12d

So we solve that system and get:

a&lt;sub&gt;1&lt;/sub&gt; = 28, d = -19/3

So to get the formula, we take

a&lt;sub&gt;n&lt;/sub&gt; = a&lt;sub&gt;1&lt;/sub&gt; + (n-1)d

And substitute those values of a&lt;sub&gt;1&lt;/sub&gt; and d

a&lt;sub&gt;n&lt;/sub&gt; = {{{28 + (n-1)(-19/3)}}}

a&lt;sub&gt;n&lt;/sub&gt; = {{{28 -(19/3)n + 19/3}}}

a&lt;sub&gt;n&lt;/sub&gt; = {{{84/3 - (19n)/3 + 19/3}}}

a&lt;sub&gt;n&lt;/sub&gt; = {{{(103-19n)/3}}}

That&#39;s the formula.  We can now fill in the blanks:

&lt;u&gt;28&lt;/u&gt;,&lt;u&gt;65/3&lt;/u&gt;,&lt;u&gt;46/3&lt;/u&gt;,&lt;u&gt;9&lt;/u&gt;,&lt;u&gt;8/3&lt;/u&gt;,&lt;u&gt;-11/3&lt;/u&gt;,&lt;u&gt;-10&lt;/u&gt;,&lt;u&gt;-49/3&lt;/u&gt;,&lt;u&gt;-68/3&lt;/u&gt;,&lt;u&gt;-29&lt;/u&gt;,&lt;u&gt;-106/3&lt;/u&gt;,&lt;u&gt;-48&lt;/u&gt;.

-------------------------------------
That&#39;s the solution your teacher wants, most likely.

However, it is also possible that it could be a 
geometric sequence as well:

Let&#39;s assume it&#39;s a geometric sequence.  Then we use

a&lt;sub&gt;n&lt;/sub&gt; = a&lt;sub&gt;1&lt;/sub&gt;r&lt;sup&gt;n-1&lt;/sup&gt;

Substitute n=7

a&lt;sub&gt;7&lt;/sub&gt; = a&lt;sub&gt;1&lt;/sub&gt;r&lt;sup&gt;7-1&lt;/sup&gt;
a&lt;sub&gt;7&lt;/sub&gt; = a&lt;sub&gt;1&lt;/sub&gt;r&lt;sup&gt;6&lt;/sup&gt;

when n=7, a&lt;sub&gt;n&lt;/sub&gt; = -10, so a&lt;sub&gt;7&lt;/sub&gt; = -10

Substituting a&lt;sub&gt;7&lt;/sub&gt; = -10

-10 = a&lt;sub&gt;1&lt;/sub&gt;r&lt;sup&gt;6&lt;/sup&gt;

Also,

a&lt;sub&gt;n&lt;/sub&gt; = a&lt;sub&gt;1&lt;/sub&gt;r&lt;sup&gt;n-1&lt;/sup&gt;

Substitute n=13

a&lt;sub&gt;13&lt;/sub&gt; = a&lt;sub&gt;1&lt;/sub&gt;r&lt;sup&gt;13-1&lt;/sup&gt;
a&lt;sub&gt;13&lt;/sub&gt; = a&lt;sub&gt;1&lt;/sub&gt;r&lt;sup&gt;12&lt;/sup&gt;

when n=13, a&lt;sub&gt;n&lt;/sub&gt; = -48, so a&lt;sub&gt;13&lt;/sub&gt; = -48

Substituting a&lt;sub&gt;13&lt;/sub&gt; = -48

-48 = a&lt;sub&gt;1&lt;/sub&gt;r&lt;sup&gt;12&lt;/sup&gt;

So we have the system of two equations and
two unknowns

-10 = a&lt;sub&gt;1&lt;/sub&gt;r&lt;sup&gt;6&lt;/sup&gt;
-48 = a&lt;sub&gt;1&lt;/sub&gt;r&lt;sup&gt;12&lt;/sup&gt;

So we solve that system and get:

a&lt;sub&gt;1&lt;/sub&gt; = {{{-25/12}}}, d = {{{root(6,(48))}}}

So to get the formula, we take

a&lt;sub&gt;n&lt;/sub&gt; = a1r&lt;sup&gt;n-1&lt;/sup&gt;

And substitute those values of a&lt;sub&gt;1&lt;/sub&gt; and d

a&lt;sub&gt;n&lt;/sub&gt; = {{{(-25/12)(root(6,48))^(n-1)}}}

That&#39;s the formula.  We can now fill in the blanks,
but I&#39;ll just give their decimal values:

&lt;u&gt;-2.083&lt;/u&gt;,&lt;u&gt;-2.706&lt;/u&gt;,&lt;u&gt;-3.514&lt;/u&gt;,&lt;u&gt;-4.564&lt;/u&gt;,&lt;u&gt;-5.928&lt;/u&gt;,&lt;u&gt;-7.699&lt;/u&gt;,&lt;u&gt;
-10&lt;/u&gt;,&lt;u&gt;-12.99&lt;/u&gt;,&lt;u&gt;-16.87&lt;/u&gt;,&lt;u&gt;-21.91&lt;/u&gt;,&lt;u&gt;-28.46&lt;/u&gt;,&lt;u&gt;-36.96&lt;/u&gt;,&lt;u&gt;-48&lt;/u&gt;.

Actually, arithmetic and geometric sequences are only 2 kinds of
many sequences.  So technically there are infinitely many answers to
this problem, but only one arithmetic sequence solution, and only one
geometric sequence solution. 

Edwin&lt;/pre&gt;</PRE>