Question 35713
<pre><font size = 4><b>40,-20,10,-5,...:t8

Four terms are listed.  Ask yourself the following questions 
about them:

What do you have to do to the 1st term to get the 2nd, that is,
What do you have to do to 40 to get -20?
Two possible answers are  (1) add -60; (2) multiply by -1/2

What do you have to do to the 2nd term to get the 3rd, that is,
What do you have to do to -20 to get 10?
Two possible answers are  (1) add 30; (2) multiply by -1/2

What do you have to do to the 3rd term to get the 4th, that is,
What do you have to do to 10 to get -5?
Two possible answers are  (1) add -15; (2) multiply by -1/2

The only answer common to all these is (2) multiply by -1/2

So it is an geometric sequence with t<sub>1</sub> = 40, and r = -1/2

t<sub>n</sub> = t<sub>1</sub>(1-r<sup>n</sup>)/(1-r) 

we are looking for t<sub>8</sub>, so n=8, t<sub>1</sub> = 40, r = -1/2

t<sub>8</sub> = 40[1-(-1/2)<sup>8</sup>]/[1-(-1/2)]

t<sub>8</sub> = 40(1-1/256)/(1+1/2)

t<sub>8</sub> = 40(255/256)/(3/2)

t<sub>8</sub> = 40(255/256)×(2/3) 

t<sub>8</sub> = 425/16 or 26.5625

Checking with a calculator:

t<sub>1</sub> = 40
t<sub>2</sub> = -20
t<sub>3</sub> = 10
t<sub>4</sub> = -5
t<sub>5</sub> = 2.5
t<sub>6</sub> = -1.25
t<sub>7</sub> = .625
t<sub>8</sub> = -.3125

Adding them all up, 40-20+10-5+2.5-1.25+.625-.3125 = 26.5625

Edwin
AnlytcPhil@aol.com</pre>