Find the Next Term 333 1/3 , 250 , 187 1/2

$333 \frac{1}{3}$ , $250$ , $187 \frac{1}{2}$

Step 1

Convert $333 \frac{1}{3}$ to an improper fraction.

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

$333 + \frac{1}{3}, 250, 187 \frac{1}{2}$

Add $333$ and $\frac{1}{3}$ .

To write $333$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$333 \cdot \frac{3}{3} + \frac{1}{3}, 250, 187 \frac{1}{2}$

Combine $333$ and $\frac{3}{3}$ .

$\frac{333 \cdot 3}{3} + \frac{1}{3}, 250, 187 \frac{1}{2}$

Combine the numerators over the common denominator.

$\frac{333 \cdot 3 + 1}{3}, 250, 187 \frac{1}{2}$

Simplify the numerator.

Multiply $333$ by $3$ .

$\frac{999 + 1}{3}, 250, 187 \frac{1}{2}$

Add $999$ and $1$ .

$\frac{1000}{3}, 250, 187 \frac{1}{2}$

Step 2

Convert $187 \frac{1}{2}$ to an improper fraction.

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

$\frac{1000}{3}, 250, 187 + \frac{1}{2}$

Add $187$ and $\frac{1}{2}$ .

To write $187$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{1000}{3}, 250, 187 \cdot \frac{2}{2} + \frac{1}{2}$

Combine $187$ and $\frac{2}{2}$ .

$\frac{1000}{3}, 250, \frac{187 \cdot 2}{2} + \frac{1}{2}$

Combine the numerators over the common denominator.

$\frac{1000}{3}, 250, \frac{187 \cdot 2 + 1}{2}$

Simplify the numerator.

Multiply $187$ by $2$ .

$\frac{1000}{3}, 250, \frac{374 + 1}{2}$

Add $374$ and $1$ .

$\frac{1000}{3}, 250, \frac{375}{2}$

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 $a_{1} = \frac{1000}{3}$ and $r = \frac{3}{4}$ .

$a_{n} = \frac{1000}{3} {(\frac{3}{4})}^{n - 1}$

Step 6

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

$a_{n} = \frac{1000}{3} \cdot \frac{3^{n - 1}}{4^{n - 1}}$

Step 7

Combine.

$a_{n} = \frac{1000 \cdot 3^{n - 1}}{3 \cdot 4^{n - 1}}$

Step 8

Cancel the common factor of $3^{n - 1}$ and $3$ .

Factor $3$ out of $1000 \cdot 3^{n - 1}$ .

$a_{n} = \frac{3 (1000 \cdot 3^{n - 2})}{3 \cdot 4^{n - 1}}$

Cancel the common factors.

Factor $3$ out of $3 \cdot 4^{n - 1}$ .

$a_{n} = \frac{3 (1000 \cdot 3^{n - 2})}{3 (4^{n - 1})}$

Cancel the common factor.

$a_{n} = \frac{3 (1000 \cdot 3^{n - 2})}{3 \cdot 4^{n - 1}}$

Rewrite the expression.

$a_{n} = \frac{1000 \cdot 3^{n - 2}}{4^{n - 1}}$

Step 9

Substitute in the value of $n$ to find the $n$ th term.

$a_{6} = \frac{1000 \cdot 3^{(6) - 2}}{4^{(6) - 1}}$

Step 10

Simplify the numerator.

Subtract $2$ from $6$ .

$a_{6} = \frac{1000 \cdot 3^{4}}{4^{6 - 1}}$

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

$a_{6} = \frac{1000 \cdot 81}{4^{6 - 1}}$

Step 11

Simplify the denominator.

Subtract $1$ from $6$ .

$a_{6} = \frac{1000 \cdot 81}{4^{5}}$

Raise $4$ to the power of $5$ .

$a_{6} = \frac{1000 \cdot 81}{1024}$

Step 12

Reduce the expression by cancelling the common factors.

Multiply $1000$ by $81$ .

$a_{6} = \frac{81000}{1024}$

Cancel the common factor of $81000$ and $1024$ .

Factor $8$ out of $81000$ .

$a_{6} = \frac{8 (10125)}{1024}$

Cancel the common factors.

Factor $8$ out of $1024$ .

$a_{6} = \frac{8 \cdot 10125}{8 \cdot 128}$

Cancel the common factor.

$a_{6} = \frac{8 \cdot 10125}{8 \cdot 128}$

Rewrite the expression.

$a_{6} = \frac{10125}{128}$

Find the Next Term 333 1/3 , 250 , 187 1/2

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