Find the Next Term 3 , 6 , 12 , 24 , 48 , 96

Solution:

Step:1

We identify the pattern in the sequence. Each term is obtained by multiplying the previous term by a constant factor. This indicates that we are dealing with a geometric sequence with a common ratio, $r$. For our sequence, the common ratio is $r = 2$.

Step:2

The general formula for the nth term of a geometric sequence is $a_{n} = a_{1} \cdot r^{n - 1}$.

Step:3

We plug in the first term of the sequence $a_{1} = 3$ and the common ratio $r = 2$ into the formula, yielding $a_{n} = 3 \cdot 2^{n - 1}$.

Step:4

To find the next term after $96$, we determine the term number of $96$ which is the 6th term, and then calculate the 7th term using $n = 7$ in our formula: $a_{7} = 3 \cdot 2^{7 - 1}$.

Step:5

We perform the subtraction in the exponent: $a_{7} = 3 \cdot 2^{6}$.

Step:6

We calculate $2$ raised to the 6th power: $a_{7} = 3 \cdot 64$.

Step:7

Finally, we multiply $3$ by $64$ to find the 7th term: $a_{7} = 192$.

Knowledge Notes:

A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. The formula for the nth term of a geometric sequence is given by:

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

where:

• $a_{n}$ is the nth term of the sequence,
• $a_{1}$ is the first term of the sequence,
• $r$ is the common ratio, and
• $n$ is the term number.

In this problem, the sequence is $3, 6, 12, 24, 48, 96$, which clearly shows that each term is twice the previous term, hence the common ratio $r$ is $2$. The first term $a_{1}$ is $3$. To find any term in the sequence, we use the formula with the appropriate values of $a_{1}$, $r$, and $n$. For instance, to find the 7th term, we use $n = 7$ and calculate $a_{7}$ accordingly.