Let $$a_1, a_2, a_3, \ldots$$ be a GP of increasing positive numbers. If the product of fourth and sixth terms is $$9$$ and the sum of fifth and seventh terms is $$24$$, then $$a_1a_9 + a_2a_4a_9 + a_5 + a_7$$ is equal to

We consider an increasing positive GP $$a_1,a_2,a_3,\ldots$$ with first term $$a$$ and common ratio $$r>1$$ satisfying the conditions $$a_4\cdot a_6=9$$ and $$a_5+a_7=24$$. Since $$a_4=ar^3$$ and $$a_6=ar^5$$, their product is $$a_4\cdot a_6=ar^3\cdot ar^5=a^2r^8=9$$, so $$(ar^4)^2=9$$ and hence $$ar^4=3$$. Also, $$a_5+a_7=ar^4+ar^6=ar^4(1+r^2)=24$$. Substituting $$ar^4=3$$ gives $$3(1+r^2)=24\implies r^2=7$$.

Using $$ar^4=3$$ and $$r^2=7$$, we find $$a=\frac{3}{r^4}=\frac{3}{49}$$.

Next, we compute the terms in the desired expression as follows:
$$a_1a_9=a\cdot ar^8=a^2r^8=(ar^4)^2=9$$
$$a_2a_4a_9=(ar)(ar^3)(ar^8)=a^3r^{12}=(ar^4)^3=27$$
while $$a_5+a_7=24$$.

Therefore,
$$a_1a_9+a_2a_4a_9+a_5+a_7=9+27+24=60$$, so the correct answer is 60.