Sissa and arithmetico-geometric sums

This problem explores the classic “Sissa” chessboard problem as a route into arithmetico‑geometric series—one of the trickier patterns you’ll meet in the HSC Sequences topic. By working through both a finite and an infinite version, you’ll learn the shift‑and‑subtract technique, a powerful trick that converts a sum with a linear term into a far simpler geometric series.

Problem Statement

• Evaluate $S_{64}=1+2+4+\cdots+2^{63}$.
• Let

$$T=\sum_{n=1}^{64} n2^{n-1}.$$

Use shift-and-subtract to find a closed form.

• Evaluate

$$\sum_{n=1}^{\infty}\frac{n}{2^n}.$$

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Hints

Use $T-2T$ for the finite AGP, and similarly $S-\frac12S$ for the infinite one.

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Solutions

The first sum is a plain geometric series with first term $1$, common ratio $2$, and $64$ terms. We apply the finite geometric series formula directly:

$$S_{64}=2^{64}-1.$$

For the finite arithmetico‑geometric sum $T$, we notice that multiplying by the common ratio $2$ shifts the powers on the geometric part and creates a cancellation when we subtract. Write out $2T$ and subtract:

$$T-2T=\sum_{n=1}^{64}2^{n-1}-64\cdot2^{64} =(2^{64}-1)-64\cdot2^{64},$$

so

$$T=63\cdot2^{64}+1.$$

The infinite series $S=\sum_{n=1}^\infty \frac{n}{2^n}$ has common ratio $\frac12$. We use the same shift‑and‑subtract idea: multiply by $\frac12$ and subtract, which turns the tail into a pure geometric series that we can sum to infinity.

$$S-\frac12S=\frac12+\frac14+\frac18+\cdots=1,$$

hence $S=2$.

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Takeaways

• A pure geometric series is just the standard formula; the real trick is handling the interfering linear factor $n$ with shift‑and‑subtract.
• Multiplying by the common ratio and subtracting aligns the series so that the arithmetic part collapses, leaving only a simple geometric series to evaluate.
• The same method works for both finite and infinite arithmetico‑geometric series, making it a universal tool for sums of the form $\sum n r^{n-1}$.

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Further Readings

HSC Differential Equations, HSC Trigonometry, HSC Collections, HSC Combinatorics