ARITHMETICAL PROGRESSION. 303
Ex. 3. The first term of a series is 5, the last 45, and the sum 400: find the number of terms, and the common difference.
If n be the number of terms, then from (1),
400=n 2 (5 + 45);
whence n = 16. If d be the common difference,
45 = the 16th term = 5 + 15 d;
whence d = 2 2 3.
EXAMPLES XXXIV. a.
1. Find the 27th and 41st terms in the series 5, 11, 17,... . 2. Find the 13th and 109th terms in the series 71, 70, 69,... . 3. Find the 17th and 54th terms in the series 10, 11 1 2, 13,... . 4. Find the 20th and 13th terms in the series —3, —2, —1,... . 5. Find the 90th and 16th terms in the series —4, 2, 5, 9,... . 6. Find the 37th and 89th terms in the series -2.8, 0, 2.8,... .
Find the last term in the following series:
7. 5, 7, 9, ... to 20 terms. 8. 7, 3, -1, ... to 15 terms. 9. 131, 9, 41, ... to 13 terms. 10. .6, 1.2, 1.8, ... to 12 terms. 11. 2.7, 3.4, 4.1, ... to 11 terms. 12. x, 2x, 3x, ... to 25 terms. 13. a — d, a + d, a + 3d,... to 30 terms. 14. 2a - b, 4a - 3b, 6a - 5b, ... to 40 terms.
Find the last term and sum of the following series:
15. 14, 64, 114, ... to 20 terms. 16. 1, 1.2, 1.4, ... to 12 terms. 17. 9, 5, 1, ... to 100 terms. 18. 1 4, -1 4, -3 4, ... to 21 terms. 19. 3 1 2, 1, -1 1 2, ... to 19 terms. 20. 64, 96, 128, ... to 16 terms.
Find the sum of the following series:
21. 5, 9, 13, ... to 19 terms. 22. 12, 9, 6, ... to 23 terms. 23. 4, 5 1 4, 6 1 2, ... to 37 terms. 24. 1O 1 2, 9, 7 1 2, ••• to 94 terms. 25. —3, 1, 5, ... to 17 terms. 26. 10, 9 2 3, 9 1 3, ... to 21 terms. 27. p, 3p, 5p,... to p terms. 28. 3a, a, —a, ... to a terms. 29. a, 0, -a, ... to a terms. 30. — 3q, — q, q, ... to p terms.
