Identify the Sequence 4 , 3 , 2 1/4 , 1 11/16

$4$ , $3$ , $2 \frac{1}{4}$ , $1 \frac{11}{16}$

Step 1

Convert $2 \frac{1}{4}$ to an improper fraction.

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

$4, 3, 2 + \frac{1}{4}, 1 \frac{11}{16}$

Add $2$ and $\frac{1}{4}$ .

To write $2$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$4, 3, 2 \cdot \frac{4}{4} + \frac{1}{4}, 1 \frac{11}{16}$

Combine $2$ and $\frac{4}{4}$ .

$4, 3, \frac{2 \cdot 4}{4} + \frac{1}{4}, 1 \frac{11}{16}$

Combine the numerators over the common denominator.

$4, 3, \frac{2 \cdot 4 + 1}{4}, 1 \frac{11}{16}$

Simplify the numerator.

Multiply $2$ by $4$ .

$4, 3, \frac{8 + 1}{4}, 1 \frac{11}{16}$

Add $8$ and $1$ .

$4, 3, \frac{9}{4}, 1 \frac{11}{16}$

Step 2

Convert $1 \frac{11}{16}$ to an improper fraction.

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

$4, 3, \frac{9}{4}, 1 + \frac{11}{16}$

Add $1$ and $\frac{11}{16}$ .

Write $1$ as a fraction with a common denominator.

$4, 3, \frac{9}{4}, \frac{16}{16} + \frac{11}{16}$

Combine the numerators over the common denominator.

$4, 3, \frac{9}{4}, \frac{16 + 11}{16}$

Add $16$ and $11$ .

$4, 3, \frac{9}{4}, \frac{27}{16}$

Step 3

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 $\frac{3}{4}$ gives the next term. In other words, $a_{n} = a_{1} r^{n - 1}$ .

Geometric Sequence: $r = \frac{3}{4}$

Step 4

This is the form of a geometric sequence.

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

Step 5

Substitute in the values of $a_{1} = 4$ and $r = \frac{3}{4}$ .

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

Step 6

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

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

Step 7

Combine $4$ and $\frac{3^{n - 1}}{4^{n - 1}}$ .

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

Identify the Sequence 4 , 3 , 2 1/4 , 1 11/16

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