Find whether each of the following numbers is a perfect square or not?
(i) 121 (ii) 55 (iii) 81
(iv) 49 (v) 69
(i) 121
∵ 121 - 1 = 120 85-13 = 72
120 - 3 = 117 72 - 15 = 57
117 - 5 = 112 54 - 17 = 40
112 - 7 = 105 40 - 19 = 21
105 - 9 = 96 21 - 21 = 0
96 - 11 = 85
i.e. 121 = 1+3+5+7+9+11+13+15+17+19+21. Thus , 121 is a perfact square.
(ii) 55
∵ 55 - 1 = 54 30 - 11 = 19
54 - 3 = 51 19 - 13 = 6
51 - 5 = 46 6 - 15 = -9
46 - 7 = 39
39 - 9 = 30
Since, 55 cannot be expressed as the sum of successive old numbers starting from 1.
∴ 55 is not a perfact square.
(iii) 81
Since, 81 - 1 = 80 56 - 11 = 45
80 - 3 = 77 45 - 13 = 32
77 - 5 = 72 32 - 15 = 17
72 - 7 = 65 17 - 17 = 0
65 - 9 =56
∴ 81 = 1+3+5+7+9+11+13+15+17
Thus, 81 is a perfact square.
(iv) 49
Since, 49 - 1 = 48
48 - 3 = 45
45 - 5 = 40
40 - 7 = 33
33 - 9 = 24
24 - 11 = 13
13 - 13 = 0
∴ 49 = 1+3+5+7+9+11+13
Thus, 69 is not a perfact square.
(v) 69
Since, 69 - 1 = 68 44 - 11 = 33
68 - 3 = 65 33 - 13 = 20
65 - 5 = 60 20 - 15 = 5
60 - 7 = 53 5 - 17 = 12
53 - 9 = 44
∴ 69 cannot be expressed as the sum of consecutive odd num numbers starting fron 1. Thus, 69 is not a perfect square.
