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the last son received $4800. How much did the first son receive? Ans $150.

3. A person bought 10 bushels of wheat, paying for it in geometrical progression, whose ratio is 3; the last bushel cost him $196.83. What did he give for the first bushel? Ans. 1 cent.

Case IV.

We also discover from Case I. that the last term divided by the first term, will give the power of the ratio, whose exponent is the number of terms, less one.

Hence, when we have given the first term, the last term, and the number of terms, to find the ratio, we have this

RULE.

Divide the last term by the first term; extract that root of the quotient which is denoted by the number of terms, less one.

EXAMPLES.

1. The first term of a geometrical progression is 1, the last term is 64, and the number of terms is 7. What is the ratio?

In this example, the last term, divided by the first term, is 64; the number of terms, less one, is 6, .. we must extract the 6th root of 64; we first extract the square root, which is 8, we now extract the cube root of 8, which is 2, for the ratio.

2. In a country, during peace, the population increased every year in the same ratio, and so fast that in the space of 5 years it became from 10000 to 14641 souls. By what ratio was the increase, yearly? Ans. .

3. The first term of a geometrical progression is 4, the last term is 78732, and the number of terms is 10. What is the ratio? Ans. 3.

Case V.

If in Case II. we write the product of the first term into the power of the ratio, whose exponent is the number of terms, less one, instead of the last term, as drawn from Case I., we shall have the sum of all the terms, repeated as many times as there are units in the number of terms, less one, equal to the power of the ratio, whose exponent is equal to the number of terms diminished by one, and multiplied by the first term.

Hence, when we have given the first term, the ratio, and the number of terms, to find the sum of all the terms, we have this

RULE.

From the power of the ratio, whose exponent is the number of terms, subtract one, divide the remainder by the ratio, less one, and multiply the quotient by the first

term.

EXAMPLES.

1. The first term of a geometrical progression is 3, the ratio is 4, and the number of terms is 9. What is the sum of all the terms?

In this example, the ratio, raised to a power whose exponent is the number of terms, is 4°262144; this, diminished by one, becomes 262143, which, divided by 3, gives 87381; this, multiplied by the first term, becomes 87381 × 3=262143, for the sum of all the terms.

2. A king in India, named SHERAN, wished (according to the Arabic author ASEPHAD,) that SESSA, the inventor of chess, should himself choose a reward. He requested the grains of wheat which arise when 1 is calculated for the first square of the board, 2 for the second square, 4 for the third, and so on; reckoning for each of the 64 squares of the board twice as many grains as for the preceding. When it was calculated, to the astonishment of the king, it was found to be an enormous number. What was it?

Ans. 18446744073709551615 grains.

3. A gentleman married his daughter on New-Year's day, and gave her husband 1 shilling towards her portion, and was to double it on the first day of every month during the year. What was her portion? Ans. £204 15 s.

Case VI.

We know from Case V. that the sum of all the terms multiplied by the ratio, less one, is equal to one subtracted from the power of the ratio, whose exponent is the number of terms, and this remainder multiplied by the first term.

Hence, when we have given the sum of all the terms, the number of terms, and the ratio, to find the first term, we have this

RULE.

Multiply the sum of all the terms by the ratio, less one; divide the product by the power of the ratio, whose index is the number of terms, after diminishing it by

one.

EXAMPLES.

1. The sum of all the terms of a geometrical progression is 262143, the number of terms is 9, and the ratio is 4. What is the first term?

In this example, the sum of all the terms, multiplied by the ratio, less one, is 262143 × 3=786429; the power of the ratio, whose exponent is the number of terms, is 4262144; this, diminished by 1, becomes 262143; .. 786429, divided by 262143, gives 3 for the first term.

2. The sum of all the terms of a geometrical progression is 5917, the number of terms is 7, and the ratio is. What is the first term?

Ans. 9.

3. If a debt of $4095 is discharged in 12 months by paying sums which are in geometrical progression, the ratio of which is 2, how much was the first payment? Ans. $1.

Case VII.

We have shown under Case II. that the sum of all the terms, multiplied by the ratio, less one, is equal to the first term subtracted from the last term into the ratio; therefore, the first term is equal to the product of the ratio into the last term, diminished by the product of the ratio, less one, into the sum of all the terms.

Hence, when we have given the sum of all the terms, the last term, and the ratio, to find the first term, we have this

RULE.

Multiply the last term by the ratio, and from the product subtract the product of the sum of all the terms into the ratio, less one.

EXAMPLES.

1. The sum of all the terms of a geometrical pro. gression is 436905, the last term is 327680, and the ratio is 4. What is the first term?

In this example, we find the last term, multiplied by the ratio, to be 1310720. The product of the sum of the terms into the ratio, less one, is 1310715; .. 1310720 —1310715=5, for the first term.

2. The sum of all the terms of a geometrical progression is 6138, the last term is 3072, and the ratio is 2. What is the first term?

Ans. 6.

3. The sum of all the terms of a geometrical progression is 1860040, the last term is 1240029, and the ratio is 3. What is the first term? Ans. 7.

Case VIII.

From the condition under Case II., we see that the ratio, multiplied into the sum of all the terms, diminished by the last term, is equal to the sum of all the terms, diminished by the first term.

Hence, when we have given the first term, the last term, and the sum of all the terms, to find the ratio, we have this

RULE.

Divide the sum of all the terms, diminished by the first term, by the sum of all the terms, diminished by the last term.

EXAMPLES.

1. The first term of a geometrical progression is 5, the last term is 327680, and the sum of all the terms is 436905. What is the ratio?

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