Can you tell me why the sum of geometric series of -3+15-75+375...,n=8 a positive number, not negative?

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Answer 1 · 2017-07-08T07:59:48

See below:

Let's look at the series this way:

We have 8 numbers. They start with #-3# and increase by multiplying by #-5#. I'm going to group these 8 numbers in groups of 2:

#-3, 15#
#-75, 375#
#-1875, 9375#
#-46875 , 234375#

If I add up each group of 2 numbers, I'll get a positive number:

#12#
#300#
#7500#
#187500#

And clearly if I add these 4 numbers up, I'll get a positive number.

Answer 2 · 2017-07-08T08:28:48

The given GP has 1st term #a=-3# and common ratio #r=-5#

We know the sum of the series up to n th term is

#S_n=(a(1-r^n))/(1-r)#,where #r=< 1#

Here #S_n=(-3(1-(-5)^n))/(1-(-5))#

#=>S_n=-1/2(1-(-5)^n)#

#=>S_n=1/2((-5)^n-1)#

#(-5)^n# is positive for all even integral value of n . So when #n=8# it will be positive and hence the sum up to 8 (even) term will be positive.