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.