General term of a geometric progression

To find the general term of a geometric progression using the formula that characterizes them, we write an expression that defines them recursively: $$$a_{n+1}=a_n \cdot r$$$

If we apply this law recursively to construct the succession, we obtain:

$$$a_2=a_1\cdot r$$$ $$$a_3=a_2\cdot r=(a_1\cdot r)\cdot r=a_1(r\cdot r)=a_1\cdot r^2$$$ $$$a_4=a_3\cdot r=(a_1\cdot r^2)\cdot r=a_1(r^2 \cdot r)=a_1\cdot r^3$$$ $$$a_5=a_4\cdot r=(a_1\cdot r^3)\cdot r=a_1(r^3 \cdot r)=a_1\cdot r^4$$$ $$$\ldots$$$

And, in general, we have $$$a_n=a_1 \cdot r^{n-1}$$$ This expression relates any term of the succession to the first by means of the ratio of the progression, that is, it is the general term of the geometric progression.

We want to find what number is in position $$37$$ in the succession $$\Big(\dfrac{1}{8},\dfrac{1}{4},\dfrac{1}{2},1,2,\ldots\Big).$$

Observe that it is a geometric progression because the ratio of two consecutive terms is constant and equal to $$2$$.

As the first term is $$a_1=\dfrac{1}{8}$$, and the ratio is $$r=2$$, we have: $$$a_n=\dfrac{2^{n-1}}{8}=\dfrac{2^{n-1}}{2^3}=2^{n-4}$$$

Since we want to find the term $$a_{37}$$, we have: $$$a_{37}=2^{37-4}=2^{33}=8.589.934.592$$$

The terms of a geometric progression can be expressed from any other term with the following expression: $$$a_m=a_k \cdot r^{m-k}$$$ since, if we apply the general term to the positions $$m$$ and $$k$$, we have: $$$a_m=a_1 \cdot r^{m-1}$$$ $$$a_k=a_1 \cdot r^{k-1}$$$

And by dividing them we obtain $$$\dfrac{a_m}{a_k}=\dfrac{a_1 \cdot r^{m-1}}{a_1 \cdot r^{k-1}}=\dfrac{r^{m-1}}{r^{k-1}}=r^{m-1-(k-1)}=r^{m-1-k+1}=r^{m-k}$$$

Then, we conclude $$$a_m=a_k \cdot r^{m-k}$$$

In a geometric progression of ratio $$r=\dfrac{1}{2}$$ in such a way that $$a_{17}=24$$, let's find the term $$a_{24}$$.

We know that $$a_m=a_k \cdot r^{m-k}$$, and so:

$$$a_{24}=a_{17}\cdot r^{24-17}=a_{17}\cdot r^7=24 \cdot \Big(\dfrac{1}{2}\Big)^7=\dfrac{24}{2^7}=\dfrac{3\cdot2^3}{2^7}=\dfrac{3}{2^4}=\dfrac{3}{16}$$$

Therefore $$a_{24}=\dfrac{3}{16}.$$