Question

Find an equation for the nth term in the following sequence:
2, 8, 32, 128, 512...

Answer 1

nth term of the sequence is
`2^(2n-1)` or
`T_(n)=2.(4)^(n-1)`

Explanation:

The given sequence is 2, 8, 32, 128, 512.........

Or we can rewrite the sequence as
`2, 2^(3), 2^(5),2^(7),2^(9) ,...............nth term.`

Now the new form of sequence confirms that the sequence is an exponential sequence.

Therefore nth term of the sequence will be
`T_(n)=2^(2n-1)` Or
`T_(n)=2.(4)^(n-1)`

Answer 2

`a_n = 2(4) ^(n-1)`

Explanation:

The sequence shown matches that of a geometric sequence of radius 4. To prove it, divide the term
`(a_(n + 1))/(a_n)` and check that
`(a_(n + 1))/(a_n)=4`

Then the formula that represents this sequence is:

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

Where
`a_1` is the first term of the series = 2 and
`r` is the radius of convergence = 4.

Then the equation is:

`a_n = 2(4) ^(n-1)`