Question

How do you find the next three terms of the sequence 96, 48, 24, 12, …?

Answer 1

In the given sequence we can see that the ratio between any two consecutive terms is equal. Hence the given sequence is in GP. Now to find the next term of the sequence we will multiply the common ratio to the last term. Hence we can find the next three terms of the sequence.

Complete step-by-step solution:
Now first let us understand the given sequences are in AP or GP.
AP means Arithmetic progression which means the difference between consecutive terms is constant. Let us say d is the common difference. Then next term is obtained by adding d to the last term.
Similarly GP means Geometric progression which means the ratio between consecutive terms is constant. If r is the common ratio between the two consecutive terms then the next term is obtained by multiplying the last term by r.
Now consider the given sequence 96, 48, 24, 12...
In the given sequence we can see that the ratio between two consecutive terms is equal.
$\dfrac{48}{96}=\dfrac{24}{48}=\dfrac{12}{24}=\dfrac{1}{2}$
Hence we can say that the given sequence is in GP where the common ratio is $\dfrac{1}{2}$ .
Now to find the next term we will multiply $\dfrac{1}{2}$ to the last term
Hence the term after 12 is given by $\dfrac{1}{2}\times 12=6$
Similarly the next term after 6 is $\dfrac{1}{2}\times 6=3$ and the term after 3 is $\dfrac{1}{2}\times 3=\dfrac{3}{2}$
Hence the next three terms of the sequence are $6,3,\dfrac{3}{2}$.

Note: Note that any GP is in the form $a,ar,a{{r}^{2}}....$ where a is the first term and r is the common ratio. In general ${{n}^{th}}$ of GP is given by ${{t}_{n}}=a{{r}^{n-1}}$ . Similarly any AP is in the form of $a,a+d,a+2d,....$and ${{n}^{th}}$ term of AP is given by ${{t}_{n}}=a+\left( n-1 \right)d$