How to find the nth term of the sequence 2, 4, 16, 256, ...?

nyungu6f5

nyungu6f5

Answered question

2023-03-10

How to find the nth term of the sequence 2 , 4 , 16 , 256 , ... ?

Answer & Explanation

enkestaxn2

enkestaxn2

Beginner2023-03-11Added 5 answers

Okay, we have 2 , 4 , 16 , 256 , ... . ?
Look for a shared distinction between each. They are all divisible by 2 . Since they are all divisible by 2 , we have 2 1 , 2 2 , 2 4 , 2 8 , ... ?
Now let's look at the power exponents, 1 , 2 , 4 , 8 , ... ?
It looks like for 1 , 2 , 4 , 8 , ... ? can work if we have 2 n , starting a 0 .
Now we have 2 2 n
Plug in to be sure:
2 2 0 = 2
2 2 1 = 4
2 2 3 = 16
2 2 4 = 256
posaminehri

posaminehri

Beginner2023-03-12Added 4 answers

Given:

2 , 4 , 16 , 256 , ...

Each element of the sequence appears to be the square of the one before it, as shown by:

4 = 2 2
16 = 4 2
256 = 16 2

This would result in the formula:

a n = 2 2 n

However, note that we have been told nothing about the nature of this sequence except the first 4 terms. We have not even been told that it is a sequence of numbers.
For example, it can be matched with a cubic formula:

a n = 1 3 ( 109 n 3 - 639 n 2 + 1160 n - 624 )

Then it would not follow the squaring pattern, but would continue:

2 , 4 , 16 , 256 , 942 , 2292 , 4524 , ...

We could choose any following numbers we like and find a formula that matches them.
No infinite sequence is determined purely by its first few terms.

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