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

1 Answer
Aviv S.
May 3, 2018

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!

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