<PRE><B>Question 810434</B>
&lt;pre&gt;

Two ways.  By listing and adding and by formula.

 19
&lt;u&gt;+17&lt;/u&gt;
 36
&lt;u&gt;+15&lt;/u&gt;
 51
&lt;u&gt;+13&lt;/u&gt;
 64
&lt;u&gt;+11&lt;/u&gt;
 75
&lt;u&gt;+ 9&lt;/u&gt;
 84
&lt;u&gt;+ 7&lt;/u&gt;
 91

Answer: 7 terms

By algebra:

This is an arithmetic series with first term = a&lt;sub&gt;1&lt;/sub&gt; = 19 and 
common difference d=-2.  We want the sum S&lt;sub&gt;n&lt;/sub&gt; to = 91, so we
substitute in

S&lt;sub&gt;n&lt;/sub&gt; = {{{n/2}}}[2a&lt;sub&gt;1&lt;/sub&gt; + (n-1)d]

and solve for n

91 = {{{n/2}}}[2(19) + (n-1)(-2)]

91 = {{{n/2}}}[2(19) + (-2)(n-1)]

91 = {{{n/2}}}[38 - 2(n-1)]

91 = {{{n/2}}}[38 - 2n + 2]

91 = {{{n/2}}}[40 - 2n]

91 = 20n - n²

n² - 20n + 91 = 0

(n-7)(n-13) = 0

n-7=0;  n-13=0
  n=7;     n=13

Answers:  7 terms and 13 terms.

13 is another solution because sooner or later
the sequence starts adding negative numbers and the sum 
starts coming back down and gets back to 91.

 19
&lt;u&gt;+17&lt;/u&gt;
 36
&lt;u&gt;+15&lt;/u&gt;
 51
&lt;u&gt;+13&lt;/u&gt;
 64
&lt;u&gt;+11&lt;/u&gt;
 75
&lt;u&gt;+ 9&lt;/u&gt;
 84
&lt;u&gt;+ 7&lt;/u&gt;
 91
&lt;u&gt;+ 5&lt;/u&gt;
 96
&lt;u&gt;+ 3&lt;/u&gt;
 99
&lt;u&gt;+ 1&lt;/u&gt;
100
&lt;u&gt; -1&lt;/u&gt;
 99
&lt;u&gt; -3&lt;/u&gt;
 96
&lt;u&gt; -5&lt;/u&gt;
 91

Edwin&lt;/pre&gt; </PRE>