Question

In a classic story, a king is asked to offer a payment as follows: place one grain of rice on one square of a chess board ( 64 squares), then two on the next square, four on the next, 8 on the next, and double the previous on each subsequent square. The king agrees, not comprehending exponential growth. But the final number (adding all the grains) is one less than $2^{64}$. How many grains is this?

Short answer

$$S_{64}={\displaystyle \frac{1(2^{64}-1)}{2-1}}=2^{64}-1$$ Thus, the total number of grains on the chessboard is $2^{64}-1$.

Step-by-step solution

Understand the doubling pattern

On the chessboard, the grains double with each square. This pattern can be written as the following series: 1, 2, 4, 8, 16, ...\

Observe the powers of 2

Notice that the doubling pattern is simply the series of powers of 2. We can rewrite the series as: $2^0$, $2^1$, $2^2$, $2^3$, ... up to $2^{63}$ (0-indexed)

Sum of the series

We need to find the sum of this series: $S=2^0+2^1+2^2+...+2^{63}$

Use the geometric series formula

This series is a geometric series with first term $a=2^0=1$ and common ratio $r=2$. The sum of a geometric series can be found using the formula $$S_n={\displaystyle \frac{a(r^n-1)}{r-1}}$$ where $S_n$ is the sum of the first $n$ terms of the series. We need to sum the first 64 terms of the series, so $n=64$. Therefore, the sum of the series is: #-}

Key concepts

Geometric Series

Understanding the geometric series is crucial when analyzing problems involving repeated multiplication or growth patterns. A geometric series is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.

For instance, in the chessboard problem involving grains of rice, each square has twice the number of grains as the square before it. This creates a geometric series with a common ratio (\r) of 2. Mathematically, a geometric series can be expressed as: $$a,ar,a{r}^2,a{r}^3,...$$, where $a$ is the first term.

To visualize:

• $a$ represents the initial quantity, which is 1 grain of rice in our example.
• $ar$ then is 2 grains of rice on the second square,
• $a{r}^2$ is 4 grains, and so on.

This pattern is the foundation of exponential growth, which occurs when a quantity increases by the same proportion in each time period.

Powers of 2

The powers of 2 are a specific type of geometric series where the common ratio is 2. This means each term of the series is twice as large as the preceding term. As each square on the chessboard corresponds to consecutive powers of 2, starting at $2^0$ (which is 1), and doubling each time, the sequence can be written as $2^0,2^1,2^2,2^3,...,2^{63}$.

Powers of 2 are ubiquitous in computer science because binary systems, which underpin all digital technology, operate on bits that are either '0' or '1', corresponding to powers of 2. Recognizing the pattern of powers of 2 can help simplify complex problems, and is also a fundamental principle seen in various disciplines such as mathematics, physics, and finance.

Sum of Series

When faced with the task of summing a geometric series, as in the case of the chessboard problem, there's a neat formula to help. For a geometric series with a first term $a$ and common ratio $r$, the sum of the first $n$ terms is given by $$S_n=\frac{a(r^n-1)}{r-1}$$.

Applying this formula to the chessboard scenario, with $a=2^0=1$ and $r=2$, and wanting the sum of 64 terms, we have: $$S_n=\frac{1(2^{64}-1)}{2-1}=2^{64}-1$$. So, the king would need $2^{64}-1$ grains of rice in total.

This concept teaches us not only a fundamental aspect of series summation but also provides a powerful insight into the nature of exponential growth and its surprising outcomes. Whether in finance with compounded interest or biology with population dynamics, the sum of series helps in calculating totals over periods, revealing the massive scales items can reach through continuous growth.