How do you find the sum of the finite geometric sequence of #Sigma 8(-1/2)^i# from i=0 to 25?

Answer 1

See below.

The sum of a geometric series is given by:

#a((1-r^n)/(1-r))#
Where #a# is the first term, #r# is the common ratio and #n# is the nth term.
For #i=0# the first term is:
#8(-1/2)^i=8(-1/2)^0=8(1)=8#
Common ratio is #-1/2#
#:.#
#8((1-(-1/2)^25)/(1-(-1/2)))=8((1-(-1/32^5))/(3/2))#
#=((16+(16/32^5))/3)=5.333333254#
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Answer 2

To find the sum of the finite geometric sequence ( \sum_{i=0}^{25} 8 \left( -\frac{1}{2} \right)^i ), you can use the formula for the sum of a finite geometric series:

[ S = \frac{a(1 - r^n)}{1 - r} ]

Where:

  • ( S ) is the sum of the series.
  • ( a ) is the first term of the series.
  • ( r ) is the common ratio.
  • ( n ) is the number of terms in the series.

For the given sequence:

  • ( a = 8 )
  • ( r = -\frac{1}{2} )
  • ( n = 26 )

Plug these values into the formula:

[ S = \frac{8(1 - (-\frac{1}{2})^{26})}{1 - (-\frac{1}{2})} ]

[ S = \frac{8(1 - \frac{1}{2^{26}})}{1 + \frac{1}{2}} ]

[ S = \frac{8(1 - \frac{1}{67108864})}{\frac{3}{2}} ]

[ S = \frac{8 \times 67108863}{3} ]

[ S = 178956970.67 ]

So, the sum of the given geometric sequence is approximately ( 178956970.67 ).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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