Pure functions instead of For loop

Question

How to get following loop output using pure functions such as NestList?

For[i = 0; v = 0, i < 4 && v < 4, i++; v++,
Print[x^i + b^v]]

and the output is

2

b+x

b^2+x^2

b^3+x^3

Answer 1

#^Range[0, 3] & /@ (x + b)

x^# + b^# &@Range[0, 3]

Answer 2

Just a generalization (of many possible):

fun[u_, n_] := Plus @@ (u^#) & /@ Range[0, n]

For example:

fun[{x,b},6]

yields:

{2, b + x, b^2 + x^2, b^3 + x^3, b^4 + x^4, b^5 + x^5, b^6 + x^6}

or

fun[{a,b,c},5]

{3, a + b + c, a^2 + b^2 + c^2, a^3 + b^3 + c^3, a^4 + b^4 + c^4, a^5 + b^5 + c^5}