In this example, the product of the common difference into the number of terms, less one, is 11×98=1078; this, added to twice the first term, gives 74+1078=1152, which, multiplied by half the number of terms, gives 57024, for the sum of all the terms.

2. The first term of an arithmetical progression is 7, the common difference is 1, and the number of terms 37. What Ans. 1258. is the sum of all the terms?

3. A person buys 37 sheep, paying for them in arithmetical progression; for the first he gives 3 shillings, and increases one shilling for each succeeding one. How much did they all come to? Ans. £38, 17s.

4. The first term of an arithmetical progression is 13, the common difference is 9, and the number of terms is 80. What is the sum of all the terms?

CASE XII.

Given the first term, the common difference, and the last term, to find the sum of all the terms.

RULE.

Divide the difference of the squares of the last and first terms, by twice the common difference, and to this quotient add half the sum of the last and first terms.

Examples.

1. The first term of an arithmetical progression is 16, the common difference is 2, and the last term 100. What is the sum of all the terms?

In this example, the difference of the squares of the last and first terms is 9744, which, divided by twice the common difference, gives 2436; this, increased by half the sum of the last and first term, becomes 2494, for the sum of all the terms,

1

2. The first term of an arithmetical progression is 5, the common difference is 7, and the last term is 75. What is the sum of all the terms? Ans. 440: 3. The first term of an arithmetical progression is 8, the common difference 3, and the last term 170. What is the sum of all the terms? Ans. 4895.

4. The first term of an arithmetical progression is 12, the common difference is 6, and the last term is 104. What is the sum of all the terms?

CASE XIII.

Given the common difference, the number of terins, and the last term, to find the sum of all the terms.

RULE.

From twice the last term, subtract the product of the common difference into the number of terms, less one; multiply this remainder by half the number of terms.

Examples.

1. The common difference of the terms of an arithmetical progression is 11, the number of terms is 19, and the last term is 199. What is the sum of all the terms?

In this example the product of the common difference into the number of terms, less one, is 11X18-198; this, subtracted from twice the last term, gives 200, which, multiplied by half the number of terms, becomes 1900, for the sum of all the terms.

2. The common difference of the terms of an arithmetical progression is 15, the number of terms is 47, and the last term is 545. What is the sum of all the terms?

Ans. 9400.

3. The common difference of the terms of an arithmetical progression is 41, the number of terms is 100, and the last term is 1000. What is the sum of all the terms?

CASE XIV.

Given the first term, the number of terms, and the sum of all the terms, to find the common difference.

RULE.

From twice the sum of the terms, subtract twice the product of the first term into the number of terms; divide this remainder by the product of the number of terms into the number of terms, less one.

Examples.

1. The first term of an arithmetical progression is 21, the number of terms is 50, and the sum of all the terms is 3500. What is the common difference?

In this example, twice the product of the first term into the number of terms is 2100; which, subtracted from twice the sum of the terms, gives 4900; the number of terms multiplied into the number of terms, less one, is 2450; hence, 4900, divided by 2450, gives 2, for the common difference.

2. The first term of an arithmetical progression is, the number of terms is 13, and the sum of all the terms is 139. What is the common difference? Ans. 18.

3. The first term of an arithmetical progression is, the number of terms is 26, and the sum of all the terms is 601. What is the common difference?

CASE XV.

Given the first term, the last term, and the sum of all the terms, to find the common difference.

RULE.

Divide the difference of the squares of the last and first terms by twice the sum of all the terms, diminished by the sum of the last and first terms.

Examples.

1. The first term of an arithmetical progression is §, the last term is 20%, and the sum of all the terms is 1394. What is the common difference?

2

In this example, the difference of the squares of the last and first terms is (204) — (4)2=4284; twice the sum of all the terms, diminished by the sum of the last and first terms, is 2574. Dividing 4284 by 2574 we get 13, for the common differ

ence.

2. The first term of an arithmetical progression is 8, the last term is 170, and the sum of all the terms is 4895. What is the common difference?

Ans. 3.

3. The first term of an arithmetical progression is 12, the last term is 104, and the sum of all the terms is 954. What is the common difference?

CASE XVI.

Given the number of terms, the last term, and the sum of all the terms, to find the common difference.

RULE.

From twice the product of the number of terms into the last term, subtract twice the sum of all the terms; divide the remainder by the product of the number of terms into the number of terms, less one.

Examples.

1. The number of terms of an arithmetical progression is 17, the last term is 50, and the sum of all the terms is 442. What is the common difference?

In this example, twice the product of the number of terms into the last term is 2× 17 × 50=1700; the product of the

number of terms into the number of terms, less one, is 17 × 16= 272: also, 1700, diminished by twice the sum of all the terms, becomes 816, which, divided by 272, gives 3, for the common difference.

2. The number of terms of an arithmetical progression is 14, the last term is 14, and the sum of all the terms is 105. What is the common difference? Ans. 1.

3. The number of terms of an arithmetical progression is 7, the last term is 41, and the sum of all the terms is 138. What is the common difference?

CASE XVII.

Given the first term, the common difference, and the sum of all the terms, to find the number of terms.

RULE.

Subtract the common difference from twice the first term, divide the remainder by twice the common difference; to the square of this quotient, add the quotient of twice the sum of all the terms, divided by the common difference; extract the square root of the sum: then divide twice the first term, diminished by the common difference by twice the common difference, and subtract this quotient from the root just found.

Examples.

1. The first term of an arithmetical progression is 7, the common difference is 1, and the sum of all the terms is 142. What is the number of terms?

In this example, the common difference, subtracted from twice the first term, gives 133, which, divided by twice the common difference, gives 27, which, squared, becomes 7561.Twice the sum of all the terms, divided by the common difference, gives 1136, which, added to 7561, gives 18921, the square