Answer:
To find the generating function for the sequence {fj}, we need to use the formula for generating functions:
G(x) = ∑fj x^j
where G(x) is the generating function.
In this case, the value fj represents the number of ways to select j pieces of fruit from a set of two apples and three bananas. We can find fj using the binomial coefficient:
fj = C(2+j-1, j) + C(3+j-1, j)
where C(n, k) is the binomial coefficient of n choose k.
Substituting this into the formula for the generating function, we get:
G(x) = ∑[C(2+j-1, j) + C(3+j-1, j)] x^j
Expanding the summation and simplifying using the binomial identity, we get:
G(x) = ∑C(1+j, j) x^j
Using the formula for the generating function of the binomial coefficient, we get:
G(x) = (1-x)^(-2)
Expanding this using the binomial series, we get:
G(x) = 1 + 2x + 3x^2 + 4x^3 + 4x^4 + 2x^5 + ...
Therefore, the correct answer is d. 1 + 2x + 3x^2 + 3x^3 + 2x^4 + x^5.