Which of the following could be used to calculate the sum of the first eight terms of the sequence whose first three terms are 6400, 800, 100, ... ?(1 point)

Question

Responses

S8=6400(1−(18)8)1−18
cap s sub 8 is equal to the fraction with numerator 6400 times open paren 1 minus 1 factorial times 8 factorial times 1 plus negative 8 factorial inverse raised to the 8 power close paren and denominator 1 minus 1 eighth

S6400=6400(1−(8)8)1−8
cap s sub 6400 is equal to the fraction with numerator 6400 times open paren 1 minus 8 to the 8th power close paren and denominator 1 minus 8

S8=8(1−(6400)18)1−6400
cap s sub 8 is equal to the fraction with numerator 8 times open paren 1 minus 6400 raised to the 1 eighth power close paren and denominator 1 minus 6400

S6400=8(1−(18)6400)1−18

Answer 1

To determine which of the provided options could be used to calculate the sum of the first eight terms of the sequence whose first three terms are 6400, 800, and 100, we first need to identify the type of sequence we are dealing with.

From the given terms:

• The first term \( a_1 = 6400 \)
• The second term \( a_2 = 800 \)
• The third term \( a_3 = 100 \)

We can check the common ratio \( r \) of the sequence: \[ r = \frac{a_2}{a_1} = \frac{800}{6400} = \frac{1}{8} \] \[ r = \frac{a_3}{a_2} = \frac{100}{800} = \frac{1}{8} \]

Since the ratio between the consecutive terms is constant, this sequence is a geometric sequence where the first term \( a = 6400 \) and the common ratio \( r = \frac{1}{8} \).

The formula for the sum of the first \( n \) terms \( S_n \) of a geometric sequence is given by: \[ S_n = a \frac{1 - r^n}{1 - r} \] For our case, where \( n = 8 \), \( a = 6400 \), and \( r = \frac{1}{8} \), the sum becomes: \[ S_8 = 6400 \frac{1 - (1/8)^8}{1 - 1/8} \]

Now, let’s explore the options you provided:

1. \( S_8 = 6400 \frac{1 - (1/8)^8}{1 - 1/8} \) (correct form)
2. \( S_{6400} = 6400 \frac{1 - 8^8}{1 - 8} \) (not in correct form for geometric series)
3. \( S_8 = 8 \frac{1 - 6400^{1/8}}{1 - 6400} \) (not a geometric series)
4. \( S_{6400} = 8 \frac{1 - (1/8)^{6400}}{1 - 1/8} \) (not in correct form)

Based on the correct formulation of the sum of a geometric series, we can conclude that here is the correct answer:

None of the provided options perfectly match the formula laid out above for \( S_8 \). However, if we apply the correct formula, we would use: \[ S_8 = 6400 \frac{1 - (1/8)^8}{1 - 1/8} \]

If one of the choices closely resembles that format (with proper ratio and terms), then it would be a suitable option.