Question

How do you find the nth term of sequence 2,4,16,256?

Answer 1

Hint: Given sequence \[2,4,16,256\]
To find the nth term we must observe the sequence, and in this question, we can observe that as every term is multiplied by its own value to get the next term, that is the first term is 2, so to get the next term we have to multiply 2 by 2 that is \[2 \times 2 = 4\]….and so on for further terms and in this way only we will find other successive terms but we need an only one-word equation to find any term that’s why will try to find it’s the nth term.

Complete step by step solution:
We can write the given sequence as, \[2,{2^2},{2^4},{2^8}\] ……..
Now to understand simply, we can rewrite the sequence as
 \[{2^{{2^{(0)}}}},{2^{{2^{(1)}}}},{2^{{2^{(2)}}}},{2^{{2^{(3)}}}}\],……...
We can observe that sequence is following a certain order
Now we can easily predict that the next term must be \[{2^{{2^{(4)}}}}\].
Sometimes it makes work difficult to find all term in a serial manner
But if we able to find its nth term then we can easily find any of its term
So, now we generalize the sequence
\[\sum\limits_{r = 1}^{r = n} {{2^{{2^{(r - 1)}}}}} \], and its nth term will be \[{2^{{2^{(n - 1)}}}}\].
Now we can easily put any value of n to find its corresponding term and hence full series.

Note: In some conditions where finding nth term is difficult we must take the ratio of successive and preceding term and then observe the relationship between them. Sometimes we also need to perform addition or subtraction to observe the pattern. In any case, our only target will be to observe the pattern.