Cube sum of a geometric series with power of two total

Expert intro: Geometric sequence formulas make the condition $a_1+a_2+\cdots+a_n=2^n-1$ very usable here, because they let us identify the common ratio and the first term from a matching pattern.

Detailed walkthrough

Recognize the geometric pattern

Let the geometric sequence be $a_1, a_2, \ldots, a_n$ with first term $a_1$ and common ratio $r$. Since the partial sum is given as

$a_1+a_2+\cdots+a_n=2^n-1,$

the expression already suggests a standard geometric-series form. A finite geometric sum is

$S_n=a_1\frac{r^n-1}{r-1} \quad (r\ne 1).$

A very natural match is obtained when $r=2$ and $a_1=1$, because then

$1+2+4+\cdots+2^{n-1}=2^n-1.$

So the sequence is

$a_k=2^{k-1} \quad \text{for } k=1,2,\ldots,n.$

That identification is the main step: once the terms are powers of 2, cubing them becomes straightforward.

Compute the cube sum directly

Each term satisfies

$a_k^3=(2^{k-1})^3=8^{k-1}.$

Therefore

$a_1^3+a_2^3+\cdots+a_n^3=1+8+8^2+\cdots+8^{n-1}.$

This is again a finite geometric series, now with first term 1 and ratio 8. Its sum is

$\frac{8^n-1}{8-1}=\frac{8^n-1}{7}.$

So the correct choice is

$\boxed{\frac{8^n-1}{7}}.$

This matches option C.

Why the answer is not a power of the original sum

A common mistake is to think the cube sum should equal $(a_1+a_2+\cdots+a_n)^3$. That is not true, because cubing a sum creates many cross terms such as $3a_ia_j(a_i+a_j)$ and $6a_1a_2a_3$ when expanded. In this problem, the terms themselves form a geometric sequence, so the cube of each term creates a new geometric sequence with ratio $r^3=8$.

Another useful perspective is to notice the pattern

$1^3+2^3+4^3+\cdots+(2^{n-1})^3,$

which is a geometric sum with ratio 8. Once the terms are written this way, the answer follows immediately from the geometric-series formula.

💡 Pitfall guide

A frequent trap in $a_1+a_2+\cdots+a_n=2^n-1$ is to treat the sum as if it uniquely determined any geometric sequence without checking the term pattern. Here the matching sequence is $1,2,4,\ldots,2^{n-1}$, not an arbitrary geometric progression with the same total. Another mistake is to cube the total sum and claim the result is $(2^n-1)^3$ or $(2^n-1)^2$; that ignores all cross terms created when a sum is cubed. A third error is mixing up the ratio of the original sequence and the ratio of the cubed sequence. The original ratio is 2, but after cubing each term the new ratio becomes 8. If you keep the exponent structure clear, the cube sum becomes a second geometric series and the computation is immediate. Always verify whether the problem asks for the sum of cubes or the cube of the sum, because those are very different expressions.

🔄 Real-world variant

If the condition were changed to $a_1+a_2+\cdots+a_n=3^n-1$ for a geometric sequence, the natural matching terms would be $1,3,9,\ldots,3^{n-1}$. Then the cube sum would become $1^3+3^3+9^3+\cdots+(3^{n-1})^3=1+27+27^2+\cdots+27^{n-1}$, which is a geometric series with ratio 27. Its sum would be $(27^n-1)/(27-1)=(27^n-1)/26$. If instead only the last term changed, such as $a_1+\cdots+a_n=2^{n+1}-1$, the sequence would shift to $1,2,4,\ldots,2^n$, and the same method would give $1+8+\cdots+8^n=(8^{n+1}-1)/7$. The method stays the same: identify the geometric pattern first, then cube term-by-term, and then apply the finite geometric-series formula to the new ratio.

🔍 Related terms

finite geometric series, sum of cubes sequence, geometric progression ratio