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THE PROGRESSIONS. 317 Ex. 3. The nth term of an A. P. is n 2 + 2: find the sum of 49 terms.

Let a be the first term, and l the last ; then by putting n = 1, and n = 49 respectively, we obtain a = ,

Ex. 4. If a, b, c, d, e be in G. P., prove that b + d is the geometric mean between a + c and c + e.

Since a, b, c, d, e are in continued proportion,

a b = b c = c d = d e

each ratio a+c b+d = b+d c+e. [Art. 347.]

Whence (b+d)2 =(a + c)(c + e).

EXAMPLES XXXIV. e.

1. Find the 6th term of the series 4, 2, 1|, •••. 2. Find the 21st term of the series 2 J, l|f, Ij^e, •••. 3. Find the 8th term of the series 1^, 1|}, 2i%, •«•. 4. Find the wth term of the series 3, 1^, 1, ... .

Find the series in which

5. The 15th term is 1 23 and the 23d term is 1 41. 6. The 2d term is 2, and the 31st term is /j. 7. The 39th term is j, and the 54th term is Jg-

Find the harmonic mean between

8. 2 and 4. 9. 1 and 13. 10. I and j. 11. - and -. 12. — ^ and — ^ 13. x + y and a: - ?/. a b X + y x — y 14. Insert two harmonic means between 4 and 12. 15. Insert three harmonic means between 2 23 and 12. 16. Insert four harmonic means between 1 and 6.