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How to Find the Partial Sum of a Geometric Sequence

When your pre-calculus teacher asks you to find the partial sum of a geometric sequence, the sum will have an upper limit and a lower limit. The common ratio of partial sums of this type has no specific restrictions.

You can find the partial sum of a geometric sequence, which has the general explicit expression of

image0.png

by using the following formula:

image1.png

For example, to find

image2.png

follow these steps:

  1. Find a1 by plugging in 1 for n.

    image3.png
  2. Find a2 by plugging in 2 for n.

    image4.png
  3. Divide a2 by a1 to find r.

    For this example, r = –3/9 = –1/3. Notice that this value is the same as the fraction in the parentheses.

    You may have noticed that 9(–1/3)n – 1 follows the general formula for

    image5.png

    (the general formula for a geometric sequence) exactly, where a1 = 9 and r = –1/3. However, if you didn't notice it, the method used in Steps 1–3 works to a tee.

  4. Plug a1, r, and k into the sum formula.

    The problem now boils down to the following simplifications:

    image6.png

    Geometric summation problems take quite a bit of work with fractions, so make sure to find a common denominator, invert, and multiply when necessary. Or you can use a calculator and then reconvert to a fraction. Just be careful to use correct parentheses when entering the numbers.

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