$\"\"$

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The given geometric series $\"\"$ .

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The sum of a geometric series is $\"\"$ ,

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Where $\"\"$ is first term and r is common ratio.

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Method 1

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$\"\"$ (Substitute $\"\"$ )

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$\"\"$ (Subtract denominator: $\"\"$ )

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$\"\"$ (Evaluate powers: $\"\"$ )

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$\"\"$ (Subtract numerator: $\"\"$ )

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Divide: $\"\"$ .

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$\"\"$ $\"\"$

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Method 2

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Find the terms by replacing n with 1, 2, 3, 4, 5, 6 and 7.

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$ (LCM is 729)

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$\"\"$ (Add: 729 + 243 + 81 + 27 + 9 + 3 + 1 = 1093)

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$\"\"$ (Multiply: $\"\"$ ) $\"\"$

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The sum of the geometric series is $\"\"$ .