# How do you find the sum of the finite geometric sequence of #Sigma 5(3/5)^n# from n=0 to 40?

12.5 (1.d.p)

First calculate the first three terms:

Find the common ratio:

The sum of a geometric sequence is:

Where a is the first term, n is the nth term and r is the common ratio.

So:

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To find the sum of the finite geometric sequence ( \Sigma 5\left(\frac{3}{5}\right)^n ) from ( n = 0 ) to ( n = 40 ), you can use the formula for the sum of a finite geometric series:

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

where: ( a ) is the first term of the sequence, ( r ) is the common ratio, and ( n ) is the number of terms.

In this sequence, ( a = 5 ) (the first term) and ( r = \frac{3}{5} ) (the common ratio).

Plugging these values into the formula and substituting ( n = 40 ), you get:

[ S_{40} = \frac{5(1 - \left(\frac{3}{5}\right)^{40+1})}{1 - \frac{3}{5}} ]

[ S_{40} = \frac{5(1 - \left(\frac{3}{5}\right)^{41})}{\frac{2}{5}} ]

[ S_{40} = 25\left(1 - \left(\frac{3}{5}\right)^{41}\right) ]

This will give you the sum of the finite geometric sequence.

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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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