1-32

1

2r

+ ; r even, }, {(− 1)13 – 3} x′′; r odd, −3 {1+(-1)='} æ”.

2(x-1) 2 (1+x2)

3

19.

cr+2
(ca) (c-b)!

(b −c) (b − a) + (c − a) (c - b) { xr.

1 1+x

xr.

21.

22.

+

1

23. (1)

x (1−x) (1+x2+1

11.

un-3un-1+3Un-2-Un-3=0; un-4un-1+6un−2 − 4un-3+Un-4=0.

12. Sn=S-2, where Σ=sum to infinity beginning with (n+1)th term. This may easily be shewn to agree with the result in Art. 325.

XXVI. PAGES 290, 291.

1. x=711t+100, y=775t +109; x=100, y=109.
2. x=519t-73, y=455t-64; x=446, y=391.
x=393t+320, y=436t+355; x=320, y=355.

3.

11. x=4, y=2, z=7.

12. x=2, y=9,
13. x=3, 7, 2, 6, 1; y=11, 4, 8, 1, 5; z=1, 1, 2, 2, 3.

z=7.

14.

x=1, 3, 2; y=5, 1, 3; z=2, 4, 3.

15.

280t+93.

16. 181, 412.

17. Denary 248, Septenary 503, Nonary 305.

18. a=11, 10, 9, 8, 6, 4, 3; b=66, 30, 18, 12, 6, 3, 2.

19.

20.

The 107th and 104th divisions, reckoning from either end.
50, 41, 35 times, excluding the first time.

1. x=7 or 1, y=4; x=7 or 5, y=6.

3. x=3, y=1, 11; x=7, y=9, 19; x=10, y=18, 22.

5. x=3, 2; y=1, 4.

4. x=2, 3, 6, 11; y=12, 7, 4, 3.
6. x=79, 27, 17, 13, 11, 9; y=157, 51, 29, 19, 13, 3.

7. x=15, y=4.

9. x=32, y=5.

8. x=170, y=39.

2. x=2, y=1.

10. x=164, y=21. 11. x=4, y=1.

12. 2x=(2+√3)"+(2−√3)"; 2√3 . y=(2+√√3)" − (2−√3)"; n being any

integer.

13. 2x=(2+√5)" +(2−√5)"; 2√5.y=(2+√5)" — (2−√√5)"; n being any even positive integer.

14. 2x=(4+17)" + (4-√17)"; 2/17. y=(4+√17)" — (4−√√/17)"; n being any odd positive integer.

The form of the answers to 15-17, 19, 20 will vary according to the mode of factorising the two sides of the equation.

21. Hendriek, Anna; Claas, Catriin; Cornelius, Geertruij.

5. n (n+1)(n+2) (n+4); n (n + 1) (n + 2) (n+3) (4n+21).