Elements of Algebra

170 ELEMENTS SECT. III. subtracting this last from 1, there remains $\frac{9}{10}$, and, if we divide the first term by this fraction, we have $\frac{1}{3}$ for the sum of the given progression. So that taking only one term of the progression, namely, $\frac{3}{10}$, the error would be $\frac{1}{30}$. And taking two terms, $\frac{3}{10} + \frac{3}{100} = \frac{33}{100}$, there would still be wanting $\frac{1}{300}$ to make the sum, which we have seen is $\frac{1}{3}$. 524. Let there now be given the infinite progression, $9 + \frac{9}{10} + \frac{9}{100} + \frac{9}{1000} + \frac{9}{10000} + \ldots$. The first term is 9, and the ratio is $\frac{1}{10}$. So that 1 minus the ratio is $\frac{9}{10}$: and $\frac{9}{\frac{9}{10}} = 10$, the sum required: which series is expressed by a decimal fraction, thus, $9.9999999, \ldots$. QUESTIONS FOR PRACTICE. 1. A servant agreed with a master to serve him eleven years without any other reward for his service than the produce of one grain of wheat for the first year; and that product to be sown the second year, and so on from year to year till the end of the time, allowing the increase to be only in a ten-fold proportion. What was the sum of the whole produce? Ans. 111111111110 grains. N. B. It is farther required, to reduce this number of grains to the proper measures of capacity, and then by supposing an average price of wheat to compute the value of the corns in money. 2. A servant agreed with a gentleman to serve him twelve months, provided he would give him a farthing for his first month's service, a penny for the second, and $4d.$ for the third, &c. What did his wages amount to? Ans. $5825l. 8s. 5\frac{1}{4}d$. 3. One Sessa, an Indian, having first invented the game of chess, shewed it to his prince, who was so delighted with it, that he promised him any reward he should ask; upon which Sessa requested that he might be allowed one grain of wheat for the first square on the chess board, two for the second, and so on, doubling continually, to 64, the whole number of squares. Now, supposing a pint to contain 7680 of those grains, and one quarter to be worth $1l. 7s. 6d.$, it is required to compute the value of the whole sum of grains. Ans. $64481488296l$.