A mixed number is an addition of its whole and fractional parts.

Add <math><mstyle displaystyle="true"><mn>10</mn></mstyle></math> and <math><mstyle displaystyle="true"><mfrac><mrow><mn>2</mn></mrow><mrow><mn>3</mn></mrow></mfrac></mstyle></math> .

To write <math><mstyle displaystyle="true"><mn>10</mn></mstyle></math> as a fraction with a common denominator, multiply by <math><mstyle displaystyle="true"><mfrac><mrow><mn>3</mn></mrow><mrow><mn>3</mn></mrow></mfrac></mstyle></math> .

Combine <math><mstyle displaystyle="true"><mn>10</mn></mstyle></math> and <math><mstyle displaystyle="true"><mfrac><mrow><mn>3</mn></mrow><mrow><mn>3</mn></mrow></mfrac></mstyle></math> .

Combine the numerators over the common denominator.

Simplify the numerator.

Multiply <math><mstyle displaystyle="true"><mn>10</mn></mstyle></math> by <math><mstyle displaystyle="true"><mn>3</mn></mstyle></math> .

Add <math><mstyle displaystyle="true"><mn>30</mn></mstyle></math> and <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> .

A mixed number is an addition of its whole and fractional parts.

Add <math><mstyle displaystyle="true"><mn>3</mn></mstyle></math> and <math><mstyle displaystyle="true"><mfrac><mrow><mn>5</mn></mrow><mrow><mn>9</mn></mrow></mfrac></mstyle></math> .

To write <math><mstyle displaystyle="true"><mn>3</mn></mstyle></math> as a fraction with a common denominator, multiply by <math><mstyle displaystyle="true"><mfrac><mrow><mn>9</mn></mrow><mrow><mn>9</mn></mrow></mfrac></mstyle></math> .

Combine <math><mstyle displaystyle="true"><mn>3</mn></mstyle></math> and <math><mstyle displaystyle="true"><mfrac><mrow><mn>9</mn></mrow><mrow><mn>9</mn></mrow></mfrac></mstyle></math> .

Combine the numerators over the common denominator.

Simplify the numerator.

Multiply <math><mstyle displaystyle="true"><mn>3</mn></mstyle></math> by <math><mstyle displaystyle="true"><mn>9</mn></mstyle></math> .

Add <math><mstyle displaystyle="true"><mn>27</mn></mstyle></math> and <math><mstyle displaystyle="true"><mn>5</mn></mstyle></math> .

A mixed number is an addition of its whole and fractional parts.

Add <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> and <math><mstyle displaystyle="true"><mfrac><mrow><mn>5</mn></mrow><mrow><mn>27</mn></mrow></mfrac></mstyle></math> .

Write <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> as a fraction with a common denominator.

Combine the numerators over the common denominator.

Add <math><mstyle displaystyle="true"><mn>27</mn></mstyle></math> and <math><mstyle displaystyle="true"><mn>5</mn></mstyle></math> .

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 <math><mstyle displaystyle="true"><mo>-</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>3</mn></mrow></mfrac></mstyle></math> gives the next term. In other words, <math><mstyle displaystyle="true"><msub><mi>a</mi><mrow><mi>n</mi></mrow></msub><mo>=</mo><msub><mi>a</mi><mrow><mn>1</mn></mrow></msub><msup><mrow><mi>r</mi></mrow><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup></mstyle></math> .

Geometric Sequence: <math><mstyle displaystyle="true"><mi>r</mi><mo>=</mo><mo>-</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>3</mn></mrow></mfrac></mstyle></math>

This is the form of a geometric sequence.

Substitute in the values of <math><mstyle displaystyle="true"><msub><mi>a</mi><mrow><mn>1</mn></mrow></msub><mo>=</mo><mo>-</mo><mn>32</mn></mstyle></math> and <math><mstyle displaystyle="true"><mi>r</mi><mo>=</mo><mo>-</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>3</mn></mrow></mfrac></mstyle></math> .

Apply the product rule to <math><mstyle displaystyle="true"><mo>-</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>3</mn></mrow></mfrac></mstyle></math> .

Apply the product rule to <math><mstyle displaystyle="true"><mfrac><mrow><mn>1</mn></mrow><mrow><mn>3</mn></mrow></mfrac></mstyle></math> .

One to any power is one.

Combine <math><mstyle displaystyle="true"><msup><mrow><mo>(</mo><mo>-</mo><mn>1</mn><mo>)</mo></mrow><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup></mstyle></math> and <math><mstyle displaystyle="true"><mfrac><mrow><mn>1</mn></mrow><mrow><msup><mrow><mn>3</mn></mrow><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup></mrow></mfrac></mstyle></math> .

Combine <math><mstyle displaystyle="true"><mo>-</mo><mn>32</mn></mstyle></math> and <math><mstyle displaystyle="true"><mfrac><mrow><msup><mrow><mo>(</mo><mo>-</mo><mn>1</mn><mo>)</mo></mrow><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup></mrow><mrow><msup><mrow><mn>3</mn></mrow><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup></mrow></mfrac></mstyle></math> .

Move the negative in front of the fraction.

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