$\"\"$

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The nth term of the geometric series is $\"\"$ .

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

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The 3rd term of the geometric series is $\"\"$ .

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

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The 6th term of the geometric series is $\"\"$ .

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

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$\"\"$ ---------------> (Equation 1)

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$\"\"$ ----------------> (Equation 2)

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Dividing the equations (2) by (1).

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$\"\"$

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$\"\"$ (Cancel common terms)

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$\"\"$ (Divide: $\"\"$ ) $\"\"$

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The power of $\"\"$ .

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$\"\"$

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Apply powers are equal property: If $\"\"$ then a = b.

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$\"\"$

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

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$\"\"$ (Evaluate power: $\"\"$ ) $\"\"$

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Apply division property of equality: a = b then $\"\"$ .

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$\"\"$ (Divide each side by 9)

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$\"\"$ (Cancel common terms)

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$\"\"$ (Divide: $\"\"$ ) $\"\"$

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

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

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$\"\"$ (Evaluate power: $\"\"$ )

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

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

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$\"\"$ (Divide: $\"\"$ ) $\"\"$

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The value of $\"\"$ .