Find the 4th Term 25 , 10 , 4

$25$ , $10$ , $4$

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 $\frac{2}{5}$ gives the next term. In other words, $a_{n} = a_{1} r^{n - 1}$ .

Geometric Sequence: $r = \frac{2}{5}$

Step 2

This is the form of a geometric sequence.

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

Step 3

Substitute in the values of $a_{1} = 25$ and $r = \frac{2}{5}$ .

$a_{n} = 25 {(\frac{2}{5})}^{n - 1}$

Step 4

Apply the product rule to $\frac{2}{5}$ .

$a_{n} = 25 \frac{2^{n - 1}}{5^{n - 1}}$

Step 5

Combine $25$ and $\frac{2^{n - 1}}{5^{n - 1}}$ .

$a_{n} = \frac{25 \cdot 2^{n - 1}}{5^{n - 1}}$

Step 6

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

$a_{4} = \frac{25 \cdot 2^{(4) - 1}}{5^{(4) - 1}}$

Step 7

Simplify the numerator.

Subtract $1$ from $4$ .

$a_{4} = \frac{25 \cdot 2^{3}}{5^{4 - 1}}$

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

$a_{4} = \frac{25 \cdot 8}{5^{4 - 1}}$

Step 8

Simplify the denominator.

Subtract $1$ from $4$ .

$a_{4} = \frac{25 \cdot 8}{5^{3}}$

Raise $5$ to the power of $3$ .

$a_{4} = \frac{25 \cdot 8}{125}$

Step 9

Reduce the expression by cancelling the common factors.

Multiply $25$ by $8$ .

$a_{4} = \frac{200}{125}$

Cancel the common factor of $200$ and $125$ .

Factor $25$ out of $200$ .

$a_{4} = \frac{25 (8)}{125}$

Cancel the common factors.

Factor $25$ out of $125$ .

$a_{4} = \frac{25 \cdot 8}{25 \cdot 5}$

Cancel the common factor.

$a_{4} = \frac{25 \cdot 8}{25 \cdot 5}$

Rewrite the expression.

$a_{4} = \frac{8}{5}$

Find the 4th Term 25 , 10 , 4

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