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Answer:
1 (1)
3 (1 + 2)
6 (1 + 2 + 3)
10 (1 + 2 + 3 + 4)
15 (1 + 2 + 3 + 4 + 5)
21 (1 + 2 + 3 + 4 + 5 + 6)
28 (1 + 2 + 3 + 4 + 5 + 6 + 7)
36 (1 + 2 + 3 + 4 + 5 + 6 + 7 + 8)
45 (1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9)
55 (1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10)
66 (1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11)
78 (1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12)
91 (1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 + 13)
105 (1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 + 13 + 14)
120 (1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 + 13 + 14 + 15)
Answer:
Here are the first 15 triangular numbers:
1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120
Step-by-step explanation:
Triangular numbers are the sum of the first
n natural numbers. The formula for the
n-th triangular number is:
=
(
+
1
)
2
T
n
=
2
n(n+1)
Here’s the step-by-step calculation for the first 15 triangular numbers:
For
=
1
n=1:
1
=
1
(
1
+
1
)
2
=
1
×
2
2
=
1
T
1
=
2
1(1+1)
=
2
1×2
=1
For
=
2
n=2:
2
=
2
(
2
+
1
)
2
=
2
×
3
2
=
3
T
2
=
2
2(2+1)
=
2
2×3
=3
For
=
3
n=3:
3
=
3
(
3
+
1
)
2
=
3
×
4
2
=
6
T
3
=
2
3(3+1)
=
2
3×4
=6
For
=
4
n=4:
4
=
4
(
4
+
1
)
2
=
4
×
5
2
=
10
T
4
=
2
4(4+1)
=
2
4×5
=10
For
=
5
n=5:
5
=
5
(
5
+
1
)
2
=
5
×
6
2
=
15
T
5
=
2
5(5+1)
=
2
5×6
=15
For
=
6
n=6:
6
=
6
(
6
+
1
)
2
=
6
×
7
2
=
21
T
6
=
2
6(6+1)
=
2
6×7
=21
For
=
7
n=7:
7
=
7
(
7
+
1
)
2
=
7
×
8
2
=
28
T
7
=
2
7(7+1)
=
2
7×8
=28
For
=
8
n=8:
8
=
8
(
8
+
1
)
2
=
8
×
9
2
=
36
T
8
=
2
8(8+1)
=
2
8×9
=36
For
=
9
n=9:
9
=
9
(
9
+
1
)
2
=
9
×
10
2
=
45
T
9
=
2
9(9+1)
=
2
9×10
=45
For
=
10
n=10:
10
=
10
(
10
+
1
)
2
=
10
×
11
2
=
55
T
10
=
2
10(10+1)
=
2
10×11
=55
For
=
11
n=11:
11
=
11
(
11
+
1
)
2
=
11
×
12
2
=
66
T
11
=
2
11(11+1)
=
2
11×12
=66
For
=
12
n=12:
12
=
12
(
12
+
1
)
2
=
12
×
13
2
=
78
T
12
=
2
12(12+1)
=
2
12×13
=78
For
=
13
n=13:
13
=
13
(
13
+
1
)
2
=
13
×
14
2
=
91
T
13
=
2
13(13+1)
=
2
13×14
=91
For
=
14
n=14:
14
=
14
(
14
+
1
)
2
=
14
×
15
2
=
105
T
14
=
2
14(14+1)
=
2
14×15
=105
For
=
15
n=15:
15
=
15
(
15
+
1
)
2
=
15
×
16
2
=
120
T
15
=
2
15(15+1)
=
2
15×16
=120
So, the first 15 triangular numbers are:
1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120