Precalculus Examples

$\sum_{n = 1}^{4} {(\frac{3}{4})}^{n}$

Step 1

Expand the series for each value of $n$ .

${(\frac{3}{4})}^{1} + {(\frac{3}{4})}^{2} + {(\frac{3}{4})}^{3} + {(\frac{3}{4})}^{4}$

Step 2

Simplify.

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Step 2.1

Simplify each term.

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Step 2.1.1

Simplify.

$\frac{3}{4} + {(\frac{3}{4})}^{2} + {(\frac{3}{4})}^{3} + {(\frac{3}{4})}^{4}$

Step 2.1.2

Apply the product rule to $\frac{3}{4}$ .

$\frac{3}{4} + \frac{3^{2}}{4^{2}} + {(\frac{3}{4})}^{3} + {(\frac{3}{4})}^{4}$

Step 2.1.3

Raise $3$ to the power of $2$ .

$\frac{3}{4} + \frac{9}{4^{2}} + {(\frac{3}{4})}^{3} + {(\frac{3}{4})}^{4}$

Step 2.1.4

Raise $4$ to the power of $2$ .

$\frac{3}{4} + \frac{9}{16} + {(\frac{3}{4})}^{3} + {(\frac{3}{4})}^{4}$

Step 2.1.5

Apply the product rule to $\frac{3}{4}$ .

$\frac{3}{4} + \frac{9}{16} + \frac{3^{3}}{4^{3}} + {(\frac{3}{4})}^{4}$

Step 2.1.6

Raise $3$ to the power of $3$ .

$\frac{3}{4} + \frac{9}{16} + \frac{27}{4^{3}} + {(\frac{3}{4})}^{4}$

Step 2.1.7

Raise $4$ to the power of $3$ .

$\frac{3}{4} + \frac{9}{16} + \frac{27}{64} + {(\frac{3}{4})}^{4}$

Step 2.1.8

Apply the product rule to $\frac{3}{4}$ .

$\frac{3}{4} + \frac{9}{16} + \frac{27}{64} + \frac{3^{4}}{4^{4}}$

Step 2.1.9

Raise $3$ to the power of $4$ .

$\frac{3}{4} + \frac{9}{16} + \frac{27}{64} + \frac{81}{4^{4}}$

Step 2.1.10

Raise $4$ to the power of $4$ .

$\frac{3}{4} + \frac{9}{16} + \frac{27}{64} + \frac{81}{256}$

Step 2.2

Find the common denominator.

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Step 2.2.1

Multiply $\frac{3}{4}$ by $\frac{64}{64}$ .

$\frac{3}{4} \cdot \frac{64}{64} + \frac{9}{16} + \frac{27}{64} + \frac{81}{256}$

Step 2.2.2

Multiply $\frac{3}{4}$ by $\frac{64}{64}$ .

$\frac{3 \cdot 64}{4 \cdot 64} + \frac{9}{16} + \frac{27}{64} + \frac{81}{256}$

Step 2.2.3

Multiply $\frac{9}{16}$ by $\frac{16}{16}$ .

$\frac{3 \cdot 64}{4 \cdot 64} + \frac{9}{16} \cdot \frac{16}{16} + \frac{27}{64} + \frac{81}{256}$

Step 2.2.4

Multiply $\frac{9}{16}$ by $\frac{16}{16}$ .

$\frac{3 \cdot 64}{4 \cdot 64} + \frac{9 \cdot 16}{16 \cdot 16} + \frac{27}{64} + \frac{81}{256}$

Step 2.2.5

Multiply $\frac{27}{64}$ by $\frac{4}{4}$ .

$\frac{3 \cdot 64}{4 \cdot 64} + \frac{9 \cdot 16}{16 \cdot 16} + \frac{27}{64} \cdot \frac{4}{4} + \frac{81}{256}$

Step 2.2.6

Multiply $\frac{27}{64}$ by $\frac{4}{4}$ .

$\frac{3 \cdot 64}{4 \cdot 64} + \frac{9 \cdot 16}{16 \cdot 16} + \frac{27 \cdot 4}{64 \cdot 4} + \frac{81}{256}$

Step 2.2.7

Multiply $4$ by $64$ .

$\frac{3 \cdot 64}{256} + \frac{9 \cdot 16}{16 \cdot 16} + \frac{27 \cdot 4}{64 \cdot 4} + \frac{81}{256}$

Step 2.2.8

Multiply $16$ by $16$ .

$\frac{3 \cdot 64}{256} + \frac{9 \cdot 16}{256} + \frac{27 \cdot 4}{64 \cdot 4} + \frac{81}{256}$

Step 2.2.9

Reorder the factors of $64 \cdot 4$ .

$\frac{3 \cdot 64}{256} + \frac{9 \cdot 16}{256} + \frac{27 \cdot 4}{4 \cdot 64} + \frac{81}{256}$

Step 2.2.10

Multiply $4$ by $64$ .

$\frac{3 \cdot 64}{256} + \frac{9 \cdot 16}{256} + \frac{27 \cdot 4}{256} + \frac{81}{256}$

Step 2.3

Combine the numerators over the common denominator.

$\frac{3 \cdot 64 + 9 \cdot 16 + 27 \cdot 4 + 81}{256}$

Step 2.4

Simplify each term.

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Step 2.4.1

Multiply $3$ by $64$ .

$\frac{192 + 9 \cdot 16 + 27 \cdot 4 + 81}{256}$

Step 2.4.2

Multiply $9$ by $16$ .

$\frac{192 + 144 + 27 \cdot 4 + 81}{256}$

Step 2.4.3

Multiply $27$ by $4$ .

$\frac{192 + 144 + 108 + 81}{256}$

Step 2.5

Add $192$ and $144$ .

$\frac{336 + 108 + 81}{256}$

Step 2.6

Add $336$ and $108$ .

$\frac{444 + 81}{256}$

Step 2.7

Add $444$ and $81$ .

$\frac{525}{256}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$\frac{525}{256}$

Decimal Form:

$2.05078125$

Mixed Number Form:

$2 \frac{13}{256}$