How do you find the sum of finite geometric series?

Answer 1

#a+ar+ar^2+\cdots+ar^n=\frac{a(r^{n+1}-1)}{r-1}# if #r!=1# and #a+a+a+\cdots+a=(n+1)a# when there are #n+1# terms (and #r=1#).

Consider a finite geometric series with #n+1# terms #a+ar+ar^2+\cdots+ar^n#. Call this sum #S_{n}#. Note that #rS_{n}=ar+ar^2+ar^3+\cdots+ar^{n+1}# so that #rS_{n}-S_{n}=ar^{n+1}-a#.
Solving this equation for #S_{n}# gives #S_{n}=\frac{a(r^{n+1}-1)}{r-1}# when #r!=1#. The number #a# is often referred to as the "first term" and the number #r# is often referred to as the "common ratio". The number #n# is the highest power of #r# in the sum and the sum itself has #n+1# terms.
If #r=1#, then #S_{n}=a+a+a+\cdots+a=(n+1)a#.
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Answer 2

To find the sum of a finite geometric series, you can use the formula:

[ S_n = a \frac{1 - r^n}{1 - r} ]

Where:

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