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3. What is the eighth term of a series whose first term is 7 and common difference 4 ?

Ans, 35. 4. What is the tenth term of a series whose first term is 9 and common difference 5?

Ans. 54. 5. What is the twelfth term of a series whose first term is 11 and common difference 6?

Ans. 77. 6. What is the twentieth term of a series whose first term is 13 and common difference 7 ?

Ans. 146. 7. What is the thirtieth term of a series whose first term is 15 and common difference 8?

Ans. 247. 8. What is the fortieth term of a series whose first term is 20 and common difference 9?

Ans. 371. 9. What is the fiftieth term of the series 1, 6, 11, 16, 21, etc.?

Ans. 246. 10. What is the hundredth term of the series 1, 7, 13, 19, 25, etc. ?.

Ans. 595. (200.) If a represent the first term of a decreasing arithmetical progression, and d the common difference, the second term of the series will be a-d, the third a-2d, the fourth a-3d, etc., and the expression for any term of the series will be

l=a-(n-1)d. Hence, to find the last term of a decreasing arith metical progression, we have the following

RULE. The last term of a decreasing arithmetical progression is equal to the first term, minus the product of the common difference into the number of terms less one.

. Examples. 1. If the first term of a decreasing progression is 80, the number of terms 15, and the common differ ence 5, what is the last term ?

Ans. l=a-(n-1)d=80–14x5=10. 2. What is the twentieth term of a series whose first term is 53 and common difference 2?

Ans. 15. 3. What is the thirtieth term of a series whose first term is 114 and common difference 3 ?

Ans. 27. 4. What is the fiftieth term of a series whose first term is 228 and common difference 4 ?

Ans. 32. 5. What is the hundredth term of a series whose first term is 648 and common difference 6?

Ans. 54.

PROBLEM II. (201.) To find the sum of the terms of an arith. metical series.

Take any arithmetical series, and under it set the same terms in an inverted order thus :

QUEST.-How may we find the last term of a decreasing arithmetic. al progression? How may we find the sum of the terms of an arith. metical series?

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Let the series be 1, 3, 5, 7, 9, 11, 13, 15; the same series in- ,

in-} 15, 13, 11, 9, 7, 5, 3, 1. verted is is The sums are 16, 16, 16, 16, 16, 16, 16, 16.

The sum of all the terms in the double series is equal to the sum of the extremes 1 and 15, repeated as many times as there are terms, that is, 8 times ; and this is double the sum of the terms of a single series. Hence the sum of the terms of the proposed series is equal to

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(202.) In order to generalize this method, put s to represent the sum of the terms of the series

a, a+d, a +2d, etc., continued to l, which we employ to represent the last term; that is,

S=a+a+d+a+2d+a+3d+ ..... tl. Under it write the same series in an inverted order thus :

S=l+i-d+1-2d+7-3d+ ...... ta. If we add together the corresponding terms of the two series, we shall obtain

2S=l+a+l+a+l+a+l+a+ ... +īta. If we represent the number of terms of the series by n, we shall have

2S=n(lta);

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RULE. The sum of the terms of an arithmetical progres. sion is equal to half the sum of the two extremes, multiplied by the number of terms.

We also see from the preceding demonstration, that the sum of the extremes is equal to the sum of any other two terms equally distant from the extremes.

Examples 1. What is the sum of the natural series of num. bers 1, 2, 3, 4, 5, etc., up to 25?

Ans. S_N((+a) 25% (25+1)

w- 2 - 2 =325. 2. The extremes of an arithmetical progression are 2 and 50, and the number of terms 17. What is the sum of the series?

Ans. 442. 3. The extremes of an arithmetical progression are 10 and 20, and the number of terms 6. What is the sum of the series?

Ans. 90. 4. The extremes of an arithmetical progression are 3 and 19, and the number of terms 9. What is the sum of the series ?

Ans. 99. 5. The extremes of an arithmetical progression aro 5 and 595, and the number of terms 60. What is the sum of the series?

Ans. 18000.

Quest.-Give the rule for finding the sum of the terms of an arith metical progression. Explain the reason of the rule.

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6. The extremes of an arithmetical progression aro 5 and 92, and the number of terms 30. What is the sum of the series ?

Ans. 1455. 7. What is the sum of 100 terms of the series 1, 3, 5, 7, 9, etc. ?

Ans. 10000. (204.) If we take the equation

l=a+(n-1)d, and transpose the term (n-1)d, we obtain

a=l-(n-1)d; that is, the first term of an increasing arithmetical progression is equal to the last term, minus the prod. uct of the common difference by the number of terms less one.

(205.) If we transpose the term a, and divide by n-1, we obtain

, I-a

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that is, in an arithmetical progression, the common difference is equal to the difference between the two extremes, divided by the number of terms less one.

Examples 1. The last term of a progression is 48, the first term 6, and the number of terms 15. What is the common difference?

Ans. 3.

Quest.-How may we find the first term of an increasing progres sion? How may we find the common difference in an arithmetica) progression ?

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