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). $\newline$ $12,24,48, \ldots$ $\newline$ Find the $7$ 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

12,24,48, \ldots

\newline

Find the

7

th term.

\newline

Answer:

Identify Pattern: Identify the pattern in the sequence. $\newline$ The given sequence is $12$ , $24$ , $48$ , which suggests that each term is double the previous term. This is a geometric sequence with a common ratio of $2$ .
Determine Terms: Determine the first term ( $a_1$ ) and the common ratio ( $r$ ) of the sequence. $\newline$ The first term $a_1$ is $12$ , and the common ratio $r$ is $2$ (since each term is multiplied by $2$ to get the next term).
Use Formula: Use the formula for the $n$ th term of a geometric sequence to find the $7$ th term. $\newline$ The $n$ th term of a geometric sequence is given by $a_n = a_1 \cdot r^{(n-1)}$ .
Substitute Values: Substitute the values of $a_1$ , $r$ , and $n$ into the formula to find the $7$ th term. $\newline$ $a_7 = 12 \times 2^{(7-1)} = 12 \times 2^6$
Calculate $7$ th Term: Calculate the value of $2^6$ and then multiply by $12$ to find the $7$ th term. $\newline$ $2^6 = 64$ $\newline$ $a_7 = 12 \times 64 = 768$