1025th term of the sequence $ 1,2,2,4,4,4,4,8,8,8,8,8,8,8,8, ... $ Consider the following sequence - $$ 1,2,2,4,4,4,4,8,8,8,8,8,8,8,8, ... $$
In this sequence, what will be the $ 1025^{th}\, term $
So, when we write down the sequence and then write the value of $ n $ (Here, $n$ stands for the number of the below term) above it  We can observe the following -

$1 - 1$
$2 - 2 $
$3 - 2$
$4 - 4$ 
$5 - 4$
. . .
$8 - 8$ 
$9 - 8$
. . .

We can notice that $ 4^{th}$ term is 4 and similarly, the $ 8^{th}$ term is 8.
So the $ 1025^{th}$ term must be 1024 as $ 1024^{th} $ term starts with 1024.
So the value of $ 1025^{th}$ term is $ 2^{10} $ . 
Is there any other method to solve this question?

Make up the frequency and cumulative frequency table:
$$\begin{array}{c|c|c}
x&f&F\\
\hline
1&1&1=2^1-1\\
2&2&3=2^2-1\\
4&4&7=2^3-1\\
8&8&15=2^4-1\\
\vdots&\vdots&\vdots\\
256&256&511=2^{9}-1\\
512&512&1023=2^{10}-1\\
1024&1024&2047=2^{11}-1\\
\vdots&\vdots&\vdots\\
2^n&2^n&2^{n+1}-1
\end{array}$$
So, your approach was efficient to notice that $a_{2^n}=2^n$. Hence, $a_{1024}=\color{red}{a_{1025}}=\cdots =a_{2047}=\color{red}{1024}$.
