13. Find 65th triangular number. *
Answers
Answer:
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Explanation:
A triangular number or triangle number counts the objects that can form an equilateral triangle. The nth triangle number is the number of dots composing a triangle with n dots on a side, and is equal to the sum of the n natural numbers from 1 to n.
The general representation of a triangular number is
Tn= 1 + 2 + 3 + 4 +...+ (n-2) + (n-1) + n,
where n is a natural number.
This sum is Tn = n * (n + 1) / 2. This is the triangular number formula to find the nth triagular number.
To prove that this formula is true, write twice the general representation and rearange the terms as below
Tn = 1 + 2 + 3 + ...+ (n-2) + (n-1) + n
Tn = n + (n-1) + (n-2) +... + 3 + 2 + 1
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Tn = (1 + n) + 2 + ( n - 1) + 3 + (n - 2) + ...+ (n-1) + 2 + n + 1
Tn = (1 + n) + (1 + n) + (1 + n) +...+ (1 + n) + (1 + n). There are n terms, so
2Tn = n * (n+1) or Tn = n * (n + 1) / 2Triangular Numbers Chart 1-100
n Δ
1 1
2 3
3 6
4 10
5 15
6 21
7 28
8 36
9 45
10 55
n Δ
11 66
12 78
13 91
14 105
15 120
16 136
17 153
18 171
19 190
20 210
n Δ
21 231
22 253
23 276
24 300
25 325
26 351
27 378
28 406
29 435
30 465
n Δ
31 496
32 528
33 561
34 595
35 630
36 666
37 703
38 741
39 780
40 820
n Δ
41 861
42 903
43 946
44 990
45 1035
46 1081
47 1128
48 1176
49 1225
50 1275
n Δ
51 1326
52 1378
53 1431
54 1485
55 1540
56 1596
57 1653
58 1711
59 1770
60 1830
n Δ
61 1891
62 1953
63 2016
64 2080
65 2145
66 2211
67 2278
68 2346
69 2415
70 2485
n Δ
71 2556
72 2628
73 2701
74 2775
75 2850
76 2926
77 3003
78 3081
79 3160
80 3240
n Δ
81 3321
82 3403
83 3486
84 3570
85 3655
86 3741
87 3828
88 3916
89 4005
90 4095
n Δ
91 4186
92 4278
93 4371
94 4465
95 4560
96 4656
97 4753
98 4851
99 4950
100 5050