Therefore, having the last term, number of terms, and sum of the series, given to find the first term, we divide twice the sum of the series by the number of terms, and subtract the last term from the quotient.

17. Let the last term be 39, number of terms 19, and the sum of the series 399, to find the first term.

268. To find the common difference, d, from the 1st and 2d equation.

We find the value of L, in the first equation, to be

L=a+(n-1)d.

Substituting this value of L for S in the 2d equation, and then transposing, we have

18. If the first term is 5, the number of terms 15, and the sum of the series 285, what is the common difference?

Ans. 2.

19. If the first term is 3, the number of terms 19, and the sum of the series 399, what is the common difference?

Ans. 2.

20. If the first term is 7, the number of terms 8, and the sum of the series 100, what is the common difference?

PROBLEMS.

Ans. 14.

1. The first term is 5, the common difference 3. What is the

7th term ?

Ans. 23.

2. The first term is 3, the common difference 41. What is the 5th term?

Ans. 201.

3. The first term is 18, the common difference. What is the 7th term ? Ans. 19.

4. The first term is 7, the common difference 21, and the number of terms 5. Required the last term. Ans. 17.

5. The first term is, the common difference . What is the 10th term? Ans. 719.

6. The first term is 0, the common difference 1. What is the 20th term? Ans. 28.

7. The first term is 10, the common difference -2. What is the 4th term? Ans. 4.

8. The first term is -8, the common difference -3. What is the 10th term?

Ans. -35.

9. The first term of a descending series is 85, common difference 7. Required the 10th term.

Ans. 22.

10. The first term is 33, the common difference 21. What is the 5th term, and the sum of the series? Ans. 12, and 39.

11. The first term in a descending series is 22, the common difference is. What is the 10th term, and the sum of the series? Ans. 1, and 133.

12. The first term is a, the common difference is d. What is the nth term? Ans. a+d(n-1).

13. What is the sum of the odd numbers from 1 to 100 ?

Ans. 2500. 14. If the first term is 41, the common difference 3, and number of terms 8, what is the sum of the series? Ans. 134.

15. If the first term is 7, the common difference -4, and the number of terms 6, what is the sum of the series?

Ans. -18.

16. If the first term is 5, the last term 19, and the number of terms 6, what are the other terms of the progression ?

Ans. 74, 103, 133, 16.

17. If the extremes are -9 and 18, and the number of terms 5, what are the other terms of the progression?

Ans. —24, 41, 114.

18. If the last term of an ascending series is 20, the common difference 5, and the number of terms 8, what is the sum of the series?

Ans. 20.

19. There is a number consisting of three digits in arithmetical progression, whose sum is 12; and, if 396 be added to the number, the digits will be inverted. What is the number? Ans. 246.

Two

20. There is a certain island 50 miles in circumference. men, A and B, set out to travel round it. A goes 10 miles each day. B goes 2 miles the first day, 5 miles the second day, and 8 miles the third day, travelling each day 3 miles further than the day preceding. How far will A and B be apart the 8th day? Ans. 30 miles.

21. John Smith and John Jones set out from Boston for the city of Washington, the distance being 440 miles. Smith started 5 days before Jones, and travels 15 miles per day. Jones travels 25 miles the first day, 23 miles the second day, and 21 miles the third day, travelling each day 2 miles less than the preceding. How far apart will Smith be from Jones at the end of the 20th day, and how far will each be from Washington?

Ans. 135 miles apart. Smith 140 miles from Washington. Jones 275 miles from Washington.

22. If the first term is, the common difference --, and the number of terms 20, what are the last term and the sum of the series? Last term, -23.

Ans.

{

Sum of the series, -21.

-12, and

23. If one extreme is, the common difference the sum of the series —14, what is the number of terms?

Ans. 12.

24. If the first term is 2, last term 24, and the sum of the

series 37, what is the number of terms?

Ans. 24.

25. If the first term be 3, the last term 17, and the number of terms 29, what are the terms of the series?

Ans. 3, 3, 4, 41, 5, 51, &c.

26. The sum of the series is 164, the number of terms 10, and the common difference, to find the first term.

Ans.

27. The first term of an arithmetical series is -5, the common difference 12; what is the 9th term?

28. What are the three means between -1 and 15?

Ans. 7.

Ans. 3, 7, and 11.

29. The first term is 14, number of terms 10, and the sum of the series 67. What is the common difference?

Ans. -.

30. There are three numbers in arithmetical progression whose sum is 10, and the product of the second and third is 33. What are those numbers? Ans. 34, 34, and 10.

31. The number of terms of an arithmetical progression is equal to the common difference, the last term is equal to 4 times the first, and the sum of the series is equal to the square of the first term. What are the series, and the sum of the series?

Ans.

~

The series, 20, 32, 44, 56, 68, 80.
Sum of the series, 300.

32. There are four numbers in arithmetical progression whose sum is 28, and the sum of whose squares is 216. What are those numbers ? Ans. 4, 6, 8, and 10.

33. Find three numbers in arithmetical progression whose sum is 9, and the sum of whose cubes is 99.

Ans. 2, 3, and 4.

34. What are those four numbers in arithmetical progression the sum of the squares of whose first two terms is 34, and the sum of the squares of the last two is 130?

Ans. 3, 5, 7, and 9.

35. A certain number consists of three digits, which are in arithmetical progression; and, if the number be divided by the sum of its digits, the quotient will be 274, but, if 396 be added

to the number, the digits will be inverted.

ber.

Required the num-
Ans. 579.

36. What are those four numbers in arithmetical progression the sum of the squares of whose extremes is 90, and the sum of the squares of the means is 74 ? Ans. 3, 5, 7, and 9.

⚫ 37. What are those four numbers in arithmetical progression whose sum is 14, and whose continued product is 120 ?

Ans. 2, 3, 4, and 5.

38. There are four numbers in arithmetical progression, the product of whose extremes is 112, and that of the means 120. What are the numbers?

39. A and B, 165 miles from design to meet. A travels one second, three the third, and so on. day, 18 the second, 16 the third, they meet?

Ans. 8, 10, 12, and 14.

each other, set out with a mile the first day, two the B travels 20 miles the first and so on. How soon will Ans. 10 days, or 33 days.

40. There are four numbers in arithmetical progression, whose continued product is 1680, and common difference is 4. Required the numbers. Ans. 14, 10, 6, 2.

The

41. Five persons undertake to reap a field of 87 acres. five terms of an arithmetical progression, whose sum is 20, will express the times in which they can severally reap an acre, and they all together can finish the job in 60 days. In how many days can each, separately, reap an acre?

Ans. 2, 3, 4, 5, 6 days.

42. A gentleman set out from Boston for New York. He travelled 25 miles the first day, 20 miles the second day, each day travelling 5 miles less than the preceding. he from Boston at the end of the eleventh day?

How far was

Ans.

43. Suppose a number of stones were laid a rod distant from each other for twenty miles, and the first stone a rod from a basket. What length of ground will that man travel over, who gathers them up singly, returning with them, one by one, to the basket? Ans. 127,591 miles, 82 rods.