13. Find the geometric mean of 4 x2 - 12x+9 and 9 x2+12x+4.

14. Show that the product of any odd number of terms of a G.P. is equal to the nth power of the middle term, n being the number of the terms.

15. Find the G.P., whose sum to infinity is 4, and whose 2d term is 3.

16. Find the G.P. whose sum to infinity is 9, and whose 2d term is 4.

17. The sum of the first 10 terms of a certain G.P. is equal to 33 times the sum of the first 5 terms. What is the common ratio?

18. If the sum of a geometrical series to infinity is n times the first term, show that the common ratio is 1

n

19. If the common ratio of successive terms of a G.P. be positive and less than, show that each term is greater than the sum of all that follow it.

20. The 6th term of a G.P. is 8 times the 3d term, and the sum of the first two terms is 24. Find the series.

21. Find the sum to infinity of the series

26. Find three numbers in G.P. such that their sum is 14, and the sum of their squares is 84.

27. The sum of the 1st, 2d, and 3d the 4th term exceeds the first by 126.

terms of a G.P. is 42, and What is the series?

28. The sum of the 1st, 2d, and 3d terms of a G.P. is to the sum of the 3d, 4th, and 5th terms as 1:4, and the 5th term is 8. What is the series?

29. Show that, if a, b, c, d be in G.P., then will a + b, b + c, c+d, and also a2 + b2, b2 + c2, c2 + d2 be in G.P.

30. Show that, if a, b, c be the pth, qth, and rth terms respectively of a G.P., then will a¬rbr−ocp−q — 1.

31. Show that, if (a2 + b2) (b2 + c2) = (ab + bc)2, then will a, b, c be in G.P.

267. Sometimes we are not told the law which connects successive terms of a series; but when a certain number of the terms are given, the law can in simple cases be at once determined.

- =

Since 936, and 27 9 18, the series is not an arithmetical progression. We then see whether it satisfies the conditions for being a geometrical progression, namely = 27, which is the case. Thus the series is a geometrical progression, whose common ratio is 3.

11. Sum the following series to 6 terms, and when possible to

12. Show that, if an odd number of quantities are in geometrical progression, the first, the middle, and the last of them are also in geometrical progression.

13. The sum of the first 7 terms of an A.P. is 49, and the sum of the next 8 terms is 176. What is the series?

14. The arithmetic mean of the 1st and 3d terms of a geometrical progression is 5 times the 2d term. Find the common

ratio.

15. If the 4th term in an arithmetical progression is the geometric mean of the 2d and 7th terms, show that the 6th term is the geometric mean of the 2d and 14th.

16. If P be the continued product of n terms of a geometrical progression whose first term is a and last term 7, show that

P2 = (al)".

17. The continued product of three numbers in G.P. is 216, and the sum of the products of them in pairs is 156. Find the numbers.

18. Divide 25 into five parts which are in A.P., and which are such that the sum of the squares of the least and greatest of them is one less than the sum of the squares of the other three.

19. Insert between 6 and 16 two numbers such that the first three may be in A.P. and the last three in G.P.

20. If a, b, c be in geometrical progression, and x, y be the arithmetic means between a, b, and b, c, respectively, prove that

CHAPTER XXVI.

HARMONICAL PROGRESSION. OTHER SIMPLE SERIES.

268. A series of quantities is said to be in Harmonical Progression when the difference between the first and the second of any three consecutive terms is to the difference between the second and the third as the first is to the third.

Thus a, b, c, d, etc., are in harmonical progression, if

and so on.

abb

c::a: c,
d::b:d,

b-c:c

269. If a, b, c are in harmonical progression, we have by definition

are in arithmetical progression.

Thus, if quantities are in harmonical progression, their reciprocals are in arithmetical progression.