12. Proof:
1 1 1 1 1 1 1
1 2 2 1 2 2 2
2 3 k 12
2 3 k k 12
1 1
2
k k 12
k 1 k
2
2
k k 1
2
k 2 k 1
2
k k 1
2
13. Proof:
1 1 1 1 1 1 1
1 2 2 1 2 2 2
2 3 k 12
2 3 k k 12
1 1
2
k k 12
k 1 k
2
2
k k 1
2
k 2 k 1
2
k k 1
2
k2 k 1
2
k k 1 k k 1
2 2
14. Proof:
1 1 1 1 1 1 1
1 2 2 1 2 2 2
2 3 k 12
2 3 k k 12
1 1
2
k k 12
k 1 k
2
2
k k 1
2
k 2 k 1
2
k k 1
2
k2 k 1
2
k k 1 k k 1
2 2
k k 1
2
k k 1
2
15. Proof:
1 1 1 1 1 1 1
1 2 2 1 2 2 2
2 3 k 12
2 3 k k 12
1 1
2
k k 12
k 1 k
2
2
k k 1
2
k 2 k 1
2
k k 1
2
k2 k 1
2
k k 1 k k 1
2 2
k k 1
2
k k 1
2
1
2
k 1
16. Proof:
1 1 1 1 1 1 1
1 2 2 1 2 2 2
2 3 k 12
2 3 k k 12
1 1
2
k k 12
k 1 k
2
2
k k 1
2
k 2 k 1
2
k k 1
2
k2 k 1
2
k k 1 k k 1
2 2
k k 1
2
k k 1
2
1
2
k 1
1 1 1 1
1 2 2 2
2 3 k 12
k 1
17. (ii) A sequence is defined by;
a1 2 an1 2 an for n 1
Show that an 2 for n 1
18. (ii) A sequence is defined by;
a1 2 an1 2 an for n 1
Show that an 2 for n 1
Test: n = 1
19. (ii) A sequence is defined by;
a1 2 an1 2 an for n 1
Show that an 2 for n 1
Test: n = 1 a1 2 2
20. (ii) A sequence is defined by;
a1 2 an1 2 an for n 1
Show that an 2 for n 1
Test: n = 1 a1 2 2
A n k ak 2
21. (ii) A sequence is defined by;
a1 2 an1 2 an for n 1
Show that an 2 for n 1
Test: n = 1 a1 2 2
A n k ak 2
P n k 1 ak 1 2
22. (ii) A sequence is defined by;
a1 2 an1 2 an for n 1
Show that an 2 for n 1
Test: n = 1 a1 2 2
A n k ak 2
P n k 1 ak 1 2
Proof:
23. (ii) A sequence is defined by;
a1 2 an1 2 an for n 1
Show that an 2 for n 1
Test: n = 1 a1 2 2
A n k ak 2
P n k 1 ak 1 2
Proof:
ak 1 2 ak
24. (ii) A sequence is defined by;
a1 2 an1 2 an for n 1
Show that an 2 for n 1
Test: n = 1 a1 2 2
A n k ak 2
P n k 1 ak 1 2
Proof:
ak 1 2 ak
22
25. (ii) A sequence is defined by;
a1 2 an1 2 an for n 1
Show that an 2 for n 1
Test: n = 1 a1 2 2
A n k ak 2
P n k 1 ak 1 2
Proof:
ak 1 2 ak
22
4
2
26. (ii) A sequence is defined by;
a1 2 an1 2 an for n 1
Show that an 2 for n 1
Test: n = 1 a1 2 2
A n k ak 2
P n k 1 ak 1 2
Proof:
ak 1 2 ak
22
4
2
ak 1 2
27. iii The sequences xn and yn are defined by;
xn yn 2x y
x1 5, y1 2 xn1 , yn1 n n
2 xn yn
Prove xn yn 10 for n 1
28. iii The sequences xn and yn are defined by;
xn yn 2x y
x1 5, y1 2 xn1 , yn1 n n
2 xn yn
Prove xn yn 10 for n 1
Test: n = 1
29. iii The sequences xn and yn are defined by;
xn yn 2x y
x1 5, y1 2 xn1 , yn1 n n
2 xn yn
Prove xn yn 10 for n 1
Test: n = 1 x1 y1 52
10
30. iii The sequences xn and yn are defined by;
xn yn 2x y
x1 5, y1 2 xn1 , yn1 n n
2 xn yn
Prove xn yn 10 for n 1
Test: n = 1 x1 y1 52
10
A n k xk yk 10
31. iii The sequences xn and yn are defined by;
xn yn 2x y
x1 5, y1 2 xn1 , yn1 n n
2 xn yn
Prove xn yn 10 for n 1
Test: n = 1 x1 y1 52
10
A n k xk yk 10
P n k 1 xk 1 yk 1 10
32. iii The sequences xn and yn are defined by;
xn yn 2x y
x1 5, y1 2 xn1 , yn1 n n
2 xn yn
Prove xn yn 10 for n 1
Test: n = 1 x1 y1 52
10
A n k xk yk 10
P n k 1 xk 1 yk 1 10
Proof:
33. iii The sequences xn and yn are defined by;
xn yn 2x y
x1 5, y1 2 xn1 , yn1 n n
2 xn yn
Prove xn yn 10 for n 1
Test: n = 1 x1 y1 52
10
A n k xk yk 10
P n k 1 xk 1 yk 1 10
Proof:
xk yk 2 xk yk
xk 1 yk 1
x y
2 k k
34. iii The sequences xn and yn are defined by;
xn yn 2x y
x1 5, y1 2 xn1 , yn1 n n
2 xn yn
Prove xn yn 10 for n 1
Test: n = 1 x1 y1 52
10
A n k xk yk 10
P n k 1 xk 1 yk 1 10
Proof:
xk yk 2 xk yk
xk 1 yk 1
x y
2 k k
xk y k
10
35. iii The sequences xn and yn are defined by;
xn yn 2x y
x1 5, y1 2 xn1 , yn1 n n
2 xn yn
Prove xn yn 10 for n 1
Test: n = 1 x1 y1 52
10
A n k xk yk 10
P n k 1 xk 1 yk 1 10
Proof:
xk yk 2 xk yk
xk 1 yk 1
x y
2 k k
xk y k
10
xk 1 yk 1 10
36. (iv) The Fibonacci sequence is defined by;
a1 a2 1 an1 an an1 for n 1
n
1 5
Prove that an for n 1
2
37. (iv) The Fibonacci sequence is defined by;
a1 a2 1 an1 an an1 for n 1
n
1 5
Prove that an for n 1
2
Test: n = 1 and n =2
38. (iv) The Fibonacci sequence is defined by;
a1 a2 1 an1 an an1 for n 1
n
1 5
Prove that an for n 1
2
Test: n = 1 and n =2
L.H .S a1
1
39. (iv) The Fibonacci sequence is defined by;
a1 a2 1 an1 an an1 for n 1
n
1 5
Prove that an for n 1
2
Test: n = 1 and n =2 1
1 5
L.H .S a1 R.H .S
2
1
1.62
40. (iv) The Fibonacci sequence is defined by;
a1 a2 1 an1 an an1 for n 1
n
1 5
Prove that an for n 1
2
Test: n = 1 and n =2 1
1 5
L.H .S a1 R.H .S
2
1
1.62
L.H .S R.H .S
41. (iv) The Fibonacci sequence is defined by;
a1 a2 1 an1 an an1 for n 1
n
1 5
Prove that an for n 1
2
Test: n = 1 and n =2 1
1 5
L.H .S a1 R.H .S
2
1
1.62
L.H .S R.H .S
L.H .S a2
1
42. (iv) The Fibonacci sequence is defined by;
a1 a2 1 an1 an an1 for n 1
n
1 5
Prove that an for n 1
2
Test: n = 1 and n =2 1
1 5
L.H .S a1 R.H .S
2
1
1.62
L.H .S R.H .S 2
1 5
L.H .S a2 R.H .S
2
1
2.62
43. (iv) The Fibonacci sequence is defined by;
a1 a2 1 an1 an an1 for n 1
n
1 5
Prove that an for n 1
2
Test: n = 1 and n =2 1
1 5
L.H .S a1 R.H .S
2
1
1.62
L.H .S R.H .S 2
1 5
L.H .S a2 R.H .S
2
1
2.62
L.H .S R.H .S
44. (iv) The Fibonacci sequence is defined by;
a1 a2 1 an1 an an1 for n 1
n
1 5
Prove that an for n 1
2
Test: n = 1 and n =2 1
1 5
L.H .S a1 R.H .S
2
1
1.62
L.H .S R.H .S 2
1 5
L.H .S a2 R.H .S
2
1
2.62
L.H .S R.H .S
k 1 k
1 5 1 5
A n k 1 & n k ak 1 & ak
2 2
45. (iv) The Fibonacci sequence is defined by;
a1 a2 1 an1 an an1 for n 1
n
1 5
Prove that an for n 1
2
Test: n = 1 and n =2 1
1 5
L.H .S a1 R.H .S
2
1
1.62
L.H .S R.H .S 2
1 5
L.H .S a2 R.H .S
2
1
2.62
L.H .S R.H .S
k 1 k
1 5 1 5
A n k 1 & n k ak 1 & ak
2 2
k 1
1 5
P n k 1 ak 1
2