To find the sum of the first seven terms of the geometric series given as 14, 21, 31.5, 47.25, ..., we first need to identify the first term \( a \) and the common ratio \( r \).
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Identify the first term: The first term \( a \) is 14.
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Identify the common ratio \( r \): We can find the common ratio by dividing the second term by the first term: \[ r = \frac{21}{14} = 1.5 \]
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Use the formula for the sum of the first \( n \) terms of a geometric series: The formula for the sum of the first \( n \) terms of a geometric series is: \[ S_n = a \frac{1 - r^n}{1 - r} \] where \( S_n \) is the sum of \( n \) terms, \( a \) is the first term, \( r \) is the common ratio, and \( n \) is the number of terms.
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Substituting values: We need to find \( S_7 \): \[ S_7 = 14 \frac{1 - (1.5)^7}{1 - 1.5} \]
Now, let's evaluate the final expression:
- The denominator \( 1 - 1.5 = -0.5 \).
- Thus, \[ S_7 = 14 \frac{1 - (1.5)^7}{-0.5} \] which simplifies to: \[ S_7 = -28(1 - (1.5)^7) \]
Now let's compare this with the given options. The correct option must have the form \( S_7 = \frac{14(1 - (1.5)^7)}{1 - 1.5} \).
Among the provided options, the correct one is:
\[ S_7 = \frac{14 - 14(1.5)^7}{1 - 1.5} \]
So the correct answer is: \[ S_7 = \frac{14 - 14(1.5)^7}{1 - 1.5} \]
This matches with the last option you provided:
- S7=14 − 14(1.5)7 / 1 − 1.5
Thus, the correct choice is:
S7=14−14(1.5)71−1.5
1 answer