The first three terms of a sequence are given. Write your answer as a decimal or whole number. Round to the nearest thousandth (if necessary). $\newline$ $6,2, \frac{2}{3}, \ldots$ $\newline$ Find the $6$ th term. $\newline$ Answer:

Full solution

Q. The first three terms of a sequence are given. Write your answer as a decimal or whole number. Round to the nearest thousandth (if necessary).

\newline

6,2, \frac{2}{3}, \ldots

\newline

Find the

6

th term.

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Answer:

Identify Pattern: Identify the pattern in the sequence. $\newline$ The sequence given is $6, 2, \frac{2}{3}, \ldots$ To find the pattern, we look at how each term is related to the previous one. We divide each term by $3$ to get the next term. $\newline$ $6 \div 3 = 2$ $\newline$ $2 \div 3 = \frac{2}{3}$ $\newline$ This is a geometric sequence with a common ratio of $\frac{1}{3}.$
Use Formula: Use the formula for the $n$ th term of a geometric sequence. $\newline$ The $n$ th term of a geometric sequence is given by the formula: $\newline$ $a_n = a_1 \cdot r^{(n-1)}$ $\newline$ where $a_n$ is the $n$ th term, $a_1$ is the first term, $r$ is the common ratio, and $n$ is the term number.
Plug in Values: Plug in the values to find the $6$ th term. $\newline$ $a_1 = 6$ (the first term) $\newline$ $r = \frac{1}{3}$ (the common ratio) $\newline$ $n = 6$ (we want to find the $6$ th term) $\newline$ $a_6 = 6 \times \left(\frac{1}{3}\right)^{6-1}$ $\newline$ $a_6 = 6 \times \left(\frac{1}{3}\right)^5$
Calculate $6$ th Term: Calculate the $6$ th term. $\newline$ $a_6 = 6 \times (1/3)^5$ $\newline$ $a_6 = 6 \times (1/243)$ $\newline$ $a_6 = 6/243$ $\newline$ $a_6 = 2/81$
Convert to Decimal: Convert the fraction to a decimal and round to the nearest thousandth if necessary. $\newline$ $\frac{2}{81}$ as a decimal is approximately $0.0247$ $\newline$ Rounded to the nearest thousandth, it is $0.025$ .