Question

Write an expression for the general term of: 3, 5/2, 9/4, 17/8,...?

Answer 1

The expression for the general term is `a_n=(2^n+1)/2^(n-1)`.

Explanation:

Here's the sequence (I rewrote `3` as `3/1`):

`3/1,5/2,9/4,17/8,cdots`

We can see that the denominators are powers of `2`:

`3/2^0,5/2^1,9/2^2,17/2^3,cdots`

We can also see that the numerators are very close to powers of two. We can rewrite them:

`(2+1)/2^0,(4+1)/2^1,(8+1)/2^2,(16+1)/2^3,cdots`

And finally, we can write the top as the sum of `1` and powers of `2`:

`(2^1+1)/2^0,(2^2+1)/2^1,(2^3+1)/2^2,(2^4+1)/2^3,cdots`

Now, we can see that the only numbers that are changing throughout the sequence are the exponents. If you list the numbers like this, it's easier to see the pattern:

`a_1=(2^1+1)/2^0`

`a_2=(2^2+1)/2^1`

`a_3=(2^3+1)/2^2`

`qquadqquadqquadquadvdots`

`a_n=(2^n+1)/2^(n-1)`

That's the expression for the general term. Hope this helped!