# Determine Sn for the geometric series? f(1)=2, r=-2, n=12

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To determine ( S_n ) for the geometric series with first term ( f(1) = 2 ), common ratio ( r = -2 ), and number of terms ( n = 12 ), we use the formula for the sum of the first ( n ) terms of a geometric series:

[ S_n = f(1) \times \frac{1 - r^n}{1 - r} ]

Substituting the given values:

[ S_{12} = 2 \times \frac{1 - (-2)^{12}}{1 - (-2)} ]

[ S_{12} = 2 \times \frac{1 - 4096}{1 + 2} ]

[ S_{12} = 2 \times \frac{-4095}{3} ]

[ S_{12} = \frac{-8190}{3} ]

[ S_{12} = -2730 ]

So, the sum of the first 12 terms of the geometric series is ( -2730 ).

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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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