fill in the blanks 5,11,25,41,55__,89
Answers
Step-by-step explanation:
What is the next number in the series: 5, 13, 25, 41, 61, 85, 113, 145, _?
5=12+22
13=22+32
25=32+42
41=42+52
61=52+62
85=62+72
113=72+82
145=82+92
181=92+102
The numbers in the series are hypotenuses of primitive Pythagorean triples. Also interesting to note between the series of the larger legs 4, 12, 24, 40, 60, 84, 112, 144,… and of hypotenuses 5, 21, 25, 41, 61, 85, 113, 145,…, there is a fascinating pattern,
32+42=52(1)
102+112+122=132+142(2)
212+222+232+242=252+262+272(3)
362+372+382+392+402=412+422+432+442(4)
One of the most interest
5 + 4 = 9 + 2*2 = 13
13 + 4 = 17 + 2*4 = 25
25 + 4 = 29 + 2*6 = 41
41 + 4 = 45 + 2* 8 = 61
61 + 4 = 65 + 2* 10 = 85
85 + 4 = 89 + 2* 12 = 113
113 + 4 = 117 + 2*14 = 145
145 + 4 = 149 + 2* 16 = 181