don't mind me, just testing some things. polite sage.
P.79: (Modern treatment, together with Ahmose's version and a bit of culture, all in a few posts):
Consider the finite geometric series [math] \displaystyle a_{1} + ... + a_{n} [/math] with the first term being equal to the common ratio of terms, or [math] \displaystyle \frac{a_{k}}{a_{k-1}} = a_{1} = 7 \;\; ; \;\; 2 \leq k \leq n [/math]
Observe that, as Chace points out, since the first term is equal to the common ratio, the following holds:
[math]
\displaystyle \sum\limits_{k=1}^{n} 7^{k} = 7 \bigg( \sum\limits_{k=1}^{n-1} 7^{k} + 1 \bigg)
[/math]
Let n = 5 and do the following:
a) Find the sum by computing the LHS by directly summing its terms, and
b) Find the same sum by computing the form inside parentheses on the RHS (state this result), and finally by multiplying by 7.
Ans, a&b): 7+49+343+2401+16807 = 7(2801) = 19607. Ahmose gives the sum when n = 5 by employing both methods, although he simply states "multiply 2801 by 7" in as many words without mentioning the derivation of 2801, which suggests some cleverness, per the above.
Now, the fun bit. :D Ahmose presents a) as a table, together with some familiar words:
[math]
\displaystyle \begin{bmatrix}
houses & 7 \\
cats & 49 \\
mice & 343 \\
spelt & 2401 \\
hekat & 16807 \\
Total & 19607 \\
\end{bmatrix}
[/math]
Even Chace cannot help making the comparison right in the midst of his text: "As I was going to Saint Ives, I met a man with seven wives. Every wife had seven sacks..." A spelt, btw, is just a thing of wheat. This will need another post to do it justice.
