1. Interpret the two G.P.s

The notation means:

• First G.P.: $2, 2^2, 2^3, \dots$ with $60$ terms.
• Second G.P.: $4, 4^2, 4^3, \dots$ with $n$ terms.

So the terms are:

$2^1,2^2,2^3,\dots,2^{60}$ and $4^1,4^2,4^3,\dots,4^n$

Since $4^k=(2^2)^k=2^{2k}$, the second G.P. contributes terms

$2^2,2^4,2^6,\dots,2^{2n}.$

2. Use the formula for geometric mean

The geometric mean of all $(60+n)$ terms is

$\left(\prod \text{all terms}\right)^{\frac{1}{60+n}}=2^{\frac{225}{8}}.$

Hence,

$\prod \text{all terms}=2^{\frac{225}{8}(60+n)}.$

Now compute the product directly.

3. Product of the first G.P.

$\prod_{k=1}^{60}2^k=2^{1+2+\cdots+60}.$

Using

$1+2+\cdots+60=\frac{60\cdot 61}{2}=1830,$

we get

$\prod_{k=1}^{60}2^k=2^{1830}.$

4. Product of the second G.P.

$\prod_{k=1}^{n}4^k=4^{1+2+\cdots+n}=4^{\frac{n(n+1)}{2}}.$

Since $4=2^2$,

$4^{\frac{n(n+1)}{2}}=(2^2)^{\frac{n(n+1)}{2}}=2^{n(n+1)}.$

5. Total product

Therefore,

$\prod \text{all terms}=2^{1830} \cdot 2^{n(n+1)}=2^{1830+n(n+1)}.$

Equating with the geometric mean expression:

$2^{1830+n(n+1)}=2^{\frac{225}{8}(60+n)}.$

So,

$1830+n(n+1)=\frac{225}{8}(60+n).$

6. Solve for $n$

First compute:

$\frac{225}{8}\cdot 60=\frac{16875}{10}?$

Let us do it carefully:

$\frac{225}{8}\cdot 60=225\cdot \frac{60}{8}=225\cdot \frac{15}{2}=\frac{3375}{2}.$

Thus,

$1830+n^2+n=\frac{3375}{2}+\frac{225n}{8}.$

Multiply by $8$:

$14640+8n^2+8n=13500+225n.$

So,

$8n^2-217n+1140=0.$

Now factorize:

$8n^2-217n+1140=(n-15)(8n-76).$

Since $8n-76=4(2n-19)$, the roots are

$n=15 \quad \text{or} \quad n=\frac{19}{2}.$

But $n$ is the number of terms, so it must be an integer. Hence,

$n=15.$

7. Evaluate $\sum_{k=1}^n k(n-k)$

We need

$\sum_{k=1}^{n} k(n-k).$

Expand:

$\sum_{k=1}^{n} k(n-k)=\sum_{k=1}^{n}(nk-k^2)=n\sum_{k=1}^{n}k-\sum_{k=1}^{n}k^2.$

For $n=15$,

$\sum_{k=1}^{15}k=\frac{15\cdot 16}{2}=120,$

$\sum_{k=1}^{15}k^2=\frac{15\cdot 16\cdot 31}{6}=1240.$

Therefore,

$\sum_{k=1}^{15}k(15-k)=15\cdot 120-1240=1800-1240=560.$

8. Check with options

The value is

$\boxed{560}$

So the correct option is:

A

9. Comparison with stored answer

Stored correct answer: C (1330)

Our derived answer is A (560). Hence the stored answer appears to be incorrect.