Hence, for finding the last term, we have the following

RULE.

I. Multiply the common difference by one less than the number of terms.

II. To the product add the first term: the sum will be the last term.

EXAMPLES.

The formula l=a+(n−1)r serves to find any term whatever, without our being obliged to determine all those which precede it.

1. If we make n=1, we have la; that is, the series will have but one term.

2. If we make n=2, we have la+r; that is, the series will have two terms, and the second term is equal to the first plus the common difference.

3. If a=
=3 and r=2, what is the 3rd term?
4. If a 5 and r=4, what is the 6th term?
5. If a=7 and r=5, what is the 9th term ?
6. If a=8 and r=5, what is the 10th term?

Ans. 7.

Ans. 25.

Ans. 47.

Ans. 53.

7. If a 20 and r-4, what is the 12th term?

Ans. 64.

8. If a 40 and r=20, what is the 50th term?

Ans. 1020.

QUEST

138. Give the rule for finding the last term of a series when the progression is increasing.

9. If a 45 and r=30, what is the 40th term?

Ans. 1215.

10. If a 30 and r=20, what is the 60th term?

Ans. 1210.

11. If a= 50 and r=10, what is the 100th term?

139. If the progression were a decreasing one, we should have

l=a-(n-1)r.

Hence, to find the last term of a decreasing progression, we have the following

RULE.

I. Multiply the common difference by one less than the number of terms.

II. Subtract the product from the first term: the remainder will be the last term.

QUEST. 139. Give the rule for finding the last term of a series, when the progression is decreasing.

EXAMPLES.

1. The first term of a decreasing progression is 60, the number of terms 20, and the common difference 3: what is the last term?

l=a-(n-1)r gives 1-60-(20-1)3=60-57-3.

2. The first term is 90, the common difference 4, and the number of terms 15: what is the last term? Ans. 34. 100, the number of terms 40, and the what is the last term?

3. The first term is common difference 2:

Ans. 22.

4. The first term is 80, the number of terms 10, and the common difference 4: what is the last term?

Ans. 44.

5. The first term is 600, the number of terms 100, and the common difference 5: what is the last term?

Ans. 105.

6. The last term is 800, the number of terms 200, and the common difference 2: what is the last term?

Ans. 402.

140. A progression by differences being given, it is proposed to prove that, the sum of any two terms, taken at equal distances from the two extremes, is equal to the sum of the two extremes.

That is, if we have the progression

is equal to the sum of the two extremes 2 and 12.

Let a b C d e.f

.

posed progression, and n the number of terms.

We will first observe that, if a denotes a term which has p terms before it, and y a term which has p terms after it, we have, from what has been said,

Which demonstrates the proposition.

Referring this proof to the previous example, if we suppose, in the first place, x to denote the second term 4, then y will denote the term 10, next to the last. If x denotes the 3rd term 6, then y will denote 8, the third term from the last.

Having proved the first part of the proposition, write the progression below itself, but in an inverse order, viz:

Calling S the sum of the terms of the first progression, 2S will be the sum of the terms in both progressions, and we shall have

2S=(a+1)+(b+k)+(c+i) ·

...

+(i+c)+(k+b)+(l+a).

Now, since all the parts a+l, b+k, c+i . to each other, and their number equal to n,

28=(a+1)n, or s = (a +1)n.

are equal

Hence, for finding the sum of an arithmetical series, we have the following

RULE.

I. Add the two extremes together, and take half their sum. II. Multiply the half-sum by the number of terms; the product will be the sum of the series.

EXAMPLES.

1. The extremes are 2 and 16, and the number of terms 8: what is the sum of the series?

2. The extremes are 3 and 27, and the number of terms 12: what is the sum of the series?

Ans. 180.

3. The extremes are 4 and 20, and the number of terms 10 what is the sum of the series?

Ans. 120.

4. The extremes are 100 and 200, and the number of terms 80 what is the sum of the series?

Ans. 12000.

5. The extremes are 500 and 60, and the number of terms 20: what is the sum of the series?

Ans. 5600.

6. The extremes are 800 and 1200, and the number of terms 50: what is the sum of the series?

Ans. 50000.

QUEST.-140. In every progression, what is the sum of the two ex. tremes equal to? What is the rule for finding the sum of an arithmeti cal series?