Precalculus Examples

- 12 - 4 - \frac{4}{3} - \dots - \frac{4}{243}

$- 12 - 4 - \frac{4}{3} - \dots - \frac{4}{243}$

Step 1

This is a geometric sequence since there is a common ratio between each term. In this case, multiplying the previous term in the sequence by

$\frac{1}{3}$ gives the next term. In other words,

$a_{n} = a_{1} r^{n - 1}$ .

Geometric Sequence:

$r = \frac{1}{3}$

Step 2

Use the formula for a geometric sequence

$a_{n} = a_{1} r^{n - 1}$ to find the number of terms,

$n$ .

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Step 2.1

Substitute the values of the first term, last term, and ratio between terms into the formula.

$\frac{4}{243} = 12 {(\frac{1}{3})}^{n - 1}$

Step 2.2

Solve for

$n$ .

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Step 2.2.1

Rewrite the equation as

$12 {(\frac{1}{3})}^{n - 1} = \frac{4}{243}$ .

$12 {(\frac{1}{3})}^{n - 1} = \frac{4}{243}$

Step 2.2.2

Divide each term in

$12 {(\frac{1}{3})}^{n - 1} = \frac{4}{243}$ by

$12$ and simplify.

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Step 2.2.2.1

Divide each term in

$12 {(\frac{1}{3})}^{n - 1} = \frac{4}{243}$ by

$12$ .

$\frac{12 {(\frac{1}{3})}^{n - 1}}{12} = \frac{\frac{4}{243}}{12}$

Step 2.2.2.2

Simplify the left side.

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Step 2.2.2.2.1

Cancel the common factor of

$12$ .

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Step 2.2.2.2.1.1

Cancel the common factor.

$\frac{12 {(\frac{1}{3})}^{n - 1}}{12} = \frac{\frac{4}{243}}{12}$

Step 2.2.2.2.1.2

Divide

${(\frac{1}{3})}^{n - 1}$ by

$1$ .

${(\frac{1}{3})}^{n - 1} = \frac{\frac{4}{243}}{12}$

Step 2.2.2.2.2

Simplify the expression.

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Step 2.2.2.2.2.1

Apply the product rule to

$\frac{1}{3}$ .

$\frac{1^{n - 1}}{3^{n - 1}} = \frac{\frac{4}{243}}{12}$

Step 2.2.2.2.2.2

One to any power is one.

$\frac{1}{3^{n - 1}} = \frac{\frac{4}{243}}{12}$

Step 2.2.2.3

Simplify the right side.

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Step 2.2.2.3.1

Multiply the numerator by the reciprocal of the denominator.

$\frac{1}{3^{n - 1}} = \frac{4}{243} \cdot \frac{1}{12}$

Step 2.2.2.3.2

Cancel the common factor of

$4$ .

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Step 2.2.2.3.2.1

Factor

$4$ out of

$12$ .

$\frac{1}{3^{n - 1}} = \frac{4}{243} \cdot \frac{1}{4 (3)}$

Step 2.2.2.3.2.2

Cancel the common factor.

$\frac{1}{3^{n - 1}} = \frac{4}{243} \cdot \frac{1}{4 \cdot 3}$

Step 2.2.2.3.2.3

Rewrite the expression.

$\frac{1}{3^{n - 1}} = \frac{1}{243} \cdot \frac{1}{3}$

Step 2.2.2.3.3

Multiply

$\frac{1}{243}$ by

$\frac{1}{3}$ .

$\frac{1}{3^{n - 1}} = \frac{1}{243 \cdot 3}$

Step 2.2.2.3.4

Multiply

$243$ by

$3$ .

$\frac{1}{3^{n - 1}} = \frac{1}{729}$

Step 2.2.3

Multiply both sides by

$3^{n - 1}$ .

$\frac{1}{3^{n - 1}} \cdot 3^{n - 1} = \frac{1}{729} \cdot 3^{n - 1}$

Step 2.2.4

Simplify.

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Step 2.2.4.1

Simplify the left side.

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Step 2.2.4.1.1

Cancel the common factor of

$3^{n - 1}$ .

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Step 2.2.4.1.1.1

Cancel the common factor.

$\frac{1}{3^{n - 1}} \cdot 3^{n - 1} = \frac{1}{729} \cdot 3^{n - 1}$

Step 2.2.4.1.1.2

Rewrite the expression.

$1 = \frac{1}{729} \cdot 3^{n - 1}$

Step 2.2.4.2

Simplify the right side.

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Step 2.2.4.2.1

Combine

$\frac{1}{729}$ and

$3^{n - 1}$ .

$1 = \frac{3^{n - 1}}{729}$

Step 2.2.5

Solve for

$n$ .

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Step 2.2.5.1

Rewrite the equation as

$\frac{3^{n - 1}}{729} = 1$ .

$\frac{3^{n - 1}}{729} = 1$

Step 2.2.5.2

Multiply both sides of the equation by

$729$ .

$729 \frac{3^{n - 1}}{729} = 729 \cdot 1$

Step 2.2.5.3

Simplify both sides of the equation.

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Step 2.2.5.3.1

Simplify the left side.

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Step 2.2.5.3.1.1

Cancel the common factor of

$729$ .

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Step 2.2.5.3.1.1.1

Cancel the common factor.

$729 \frac{3^{n - 1}}{729} = 729 \cdot 1$

Step 2.2.5.3.1.1.2

Rewrite the expression.

$3^{n - 1} = 729 \cdot 1$

Step 2.2.5.3.2

Simplify the right side.

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Step 2.2.5.3.2.1

Multiply

$729$ by

$1$ .

$3^{n - 1} = 729$

Step 2.2.5.4

Create equivalent expressions in the equation that all have equal bases.

$3^{n - 1} = 3^{6}$

Step 2.2.5.5

Since the bases are the same, then two expressions are only equal if the exponents are also equal.

$n - 1 = 6$

Step 2.2.5.6

Move all terms not containing

$n$ to the right side of the equation.

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Step 2.2.5.6.1

Add

$1$ to both sides of the equation.

$n = 6 + 1$

Step 2.2.5.6.2

Add

$6$ and

$1$ .

$n = 7$

Step 3

Use the formula for the sum of a geometric sequence

$S_{n} = \frac{a_{1} (r^{n} - 1)}{r - 1}$ to find the sum.

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Step 3.1

Substitute the values of the first term, ratio, and the number of terms into the sum formula.

$S_{n} = \frac{- 12 ({(\frac{1}{3})}^{7} - 1)}{\frac{1}{3} - 1}$

Step 3.2

Simplify.

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Step 3.2.1

Simplify the numerator.

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Step 3.2.1.1

Apply the product rule to

$\frac{1}{3}$ .

$S_{n} = \frac{- 12 (\frac{1^{7}}{3^{7}} - 1)}{\frac{1}{3} - 1}$

Step 3.2.1.2

One to any power is one.

$S_{n} = \frac{- 12 (\frac{1}{3^{7}} - 1)}{\frac{1}{3} - 1}$

Step 3.2.1.3

Raise

$3$ to the power of

$7$ .

$S_{n} = \frac{- 12 (\frac{1}{2187} - 1)}{\frac{1}{3} - 1}$

Step 3.2.1.4

To write

$- 1$ as a fraction with a common denominator, multiply by

$\frac{2187}{2187}$ .

$S_{n} = \frac{- 12 (\frac{1}{2187} - 1 \cdot \frac{2187}{2187})}{\frac{1}{3} - 1}$

Step 3.2.1.5

Combine

$- 1$ and

$\frac{2187}{2187}$ .

$S_{n} = \frac{- 12 (\frac{1}{2187} + \frac{- 1 \cdot 2187}{2187})}{\frac{1}{3} - 1}$

Step 3.2.1.6

Combine the numerators over the common denominator.

$S_{n} = \frac{- 12 \frac{1 - 1 \cdot 2187}{2187}}{\frac{1}{3} - 1}$

Step 3.2.1.7

Simplify the numerator.

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Step 3.2.1.7.1

Multiply

$- 1$ by

$2187$ .

$S_{n} = \frac{- 12 \frac{1 - 2187}{2187}}{\frac{1}{3} - 1}$

Step 3.2.1.7.2

Subtract

$2187$ from

$1$ .

$S_{n} = \frac{- 12 (\frac{- 2186}{2187})}{\frac{1}{3} - 1}$

Step 3.2.1.8

Move the negative in front of the fraction.

$S_{n} = \frac{- 12 \cdot - 1 \frac{2186}{2187}}{\frac{1}{3} - 1}$

Step 3.2.1.9

Combine exponents.

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Step 3.2.1.9.1

Factor out negative.

$S_{n} = \frac{- (- 12 (\frac{2186}{2187}))}{\frac{1}{3} - 1}$

Step 3.2.1.9.2

Combine

$- 12$ and

$\frac{2186}{2187}$ .

$S_{n} = \frac{- \frac{- 12 \cdot 2186}{2187}}{\frac{1}{3} - 1}$

Step 3.2.1.9.3

Multiply

$- 12$ by

$2186$ .

$S_{n} = \frac{- \frac{- 26232}{2187}}{\frac{1}{3} - 1}$

Step 3.2.1.10

Cancel the common factor of

$- 26232$ and

$2187$ .

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Step 3.2.1.10.1

Factor

$3$ out of

$- 26232$ .

$S_{n} = \frac{- \frac{3 (- 8744)}{2187}}{\frac{1}{3} - 1}$

Step 3.2.1.10.2

Cancel the common factors.

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Step 3.2.1.10.2.1

Factor

$3$ out of

$2187$ .

$S_{n} = \frac{- \frac{3 \cdot - 8744}{3 \cdot 729}}{\frac{1}{3} - 1}$

Step 3.2.1.10.2.2

Cancel the common factor.

$S_{n} = \frac{- \frac{3 \cdot - 8744}{3 \cdot 729}}{\frac{1}{3} - 1}$

Step 3.2.1.10.2.3

Rewrite the expression.

$S_{n} = \frac{- \frac{- 8744}{729}}{\frac{1}{3} - 1}$

Step 3.2.1.11

Move the negative in front of the fraction.

$S_{n} = \frac{- - 1 \frac{8744}{729}}{\frac{1}{3} - 1}$

Step 3.2.1.12

Combine exponents.

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Step 3.2.1.12.1

Factor out negative.

$S_{n} = \frac{- - \frac{8744}{729}}{\frac{1}{3} - 1}$

Step 3.2.1.12.2

Multiply

$- 1$ by

$- 1$ .

$S_{n} = \frac{1 (\frac{8744}{729})}{\frac{1}{3} - 1}$

Step 3.2.1.12.3

Multiply

$\frac{8744}{729}$ by

$1$ .

$S_{n} = \frac{\frac{8744}{729}}{\frac{1}{3} - 1}$

Step 3.2.2

Simplify the denominator.

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Step 3.2.2.1

To write

$- 1$ as a fraction with a common denominator, multiply by

$\frac{3}{3}$ .

$S_{n} = \frac{\frac{8744}{729}}{\frac{1}{3} - 1 \cdot \frac{3}{3}}$

Step 3.2.2.2

Combine

$- 1$ and

$\frac{3}{3}$ .

$S_{n} = \frac{\frac{8744}{729}}{\frac{1}{3} + \frac{- 1 \cdot 3}{3}}$

Step 3.2.2.3

Combine the numerators over the common denominator.

$S_{n} = \frac{\frac{8744}{729}}{\frac{1 - 1 \cdot 3}{3}}$

Step 3.2.2.4

Simplify the numerator.

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Step 3.2.2.4.1

Multiply

$- 1$ by

$3$ .

$S_{n} = \frac{\frac{8744}{729}}{\frac{1 - 3}{3}}$

Step 3.2.2.4.2

Subtract

$3$ from

$1$ .

$S_{n} = \frac{\frac{8744}{729}}{\frac{- 2}{3}}$

Step 3.2.2.5

Move the negative in front of the fraction.

$S_{n} = \frac{\frac{8744}{729}}{- \frac{2}{3}}$

Step 3.2.3

Multiply the numerator by the reciprocal of the denominator.

$S_{n} = \frac{8744}{729} (- \frac{3}{2})$

Step 3.2.4

Cancel the common factor of

$2$ .

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Step 3.2.4.1

Move the leading negative in

$- \frac{3}{2}$ into the numerator.

$S_{n} = \frac{8744}{729} \cdot \frac{- 3}{2}$

Step 3.2.4.2

Factor

$2$ out of

$8744$ .

$S_{n} = \frac{2 (4372)}{729} \cdot \frac{- 3}{2}$

Step 3.2.4.3

Cancel the common factor.

$S_{n} = \frac{2 \cdot 4372}{729} \cdot \frac{- 3}{2}$

Step 3.2.4.4

Rewrite the expression.

$S_{n} = \frac{4372}{729} \cdot - 3$

Step 3.2.5

Cancel the common factor of

$3$ .

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Step 3.2.5.1

Factor

$3$ out of

$729$ .

$S_{n} = \frac{4372}{3 (243)} \cdot - 3$

Step 3.2.5.2

Factor

$3$ out of

$- 3$ .

$S_{n} = \frac{4372}{3 \cdot 243} \cdot (3 \cdot - 1)$

Step 3.2.5.3

Cancel the common factor.

$S_{n} = \frac{4372}{3 \cdot 243} \cdot (3 \cdot - 1)$

Step 3.2.5.4

Rewrite the expression.

$S_{n} = \frac{4372}{243} \cdot - 1$

Step 3.2.6

Combine

$\frac{4372}{243}$ and

$- 1$ .

$S_{n} = \frac{4372 \cdot - 1}{243}$

Step 3.2.7

Multiply

$4372$ by

$- 1$ .

$S_{n} = \frac{- 4372}{243}$

Step 3.2.8

Move the negative in front of the fraction.

$S_{n} = - \frac{4372}{243}$