Question

In the geometric progression 1, 2, 4… what term is 512?

Answer 1

The next term is double the previous term, so that the
`n`-th term is given recursively by

`\begin{cases}a_1=1\\a_n=2a_(n-1)&\text{for }n>1\end{cases}`

This rule tells us that

`a_2=2a_1`

`a_3=2a_2=2^2a_1`

`a_4=2a_3=2^3a_1`

and so on, with the explicit rule

`a_n=2^(n-1)a_1=2^(n-1)`

for
`n\ge1`.

If 512 is the
`k`-th term in the sequence, then

`512=2^(k-1)\implies\log_2512=\log_22^9=\log_22^(k-1)\implies9=k-1\implies k=10`