Two Famous Formulae
1 3 + 2 3 + ⋯ + n 3 = 1 + 2 + ⋯ + n
For all positive integers n , is the above equation true or false?
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3 solutions
YES .IT'S IS MY ANSWER.
Did the same Thing
This can actually be illustrated with the use of a simple multiplication table, that children use to learn how to multiply. Here is a multiplication table
Notice that given any n × n array of numbers with 1 at the upper left corner, the sum of all the numbers in the array equals the square of the sum of all the numbers in the top row. For example, for n = 4 , we have
1 + 2 + 3 + 4 = 1 0
1
+
2
+
3
+
4
+
2
+
4
+
6
+
8
+
3
+
6
+
9
+
1
2
+
4
+
8
+
1
2
+
1
6
=
1
0
0
which is the sum of the first n cubes
1 + 8 + 2 7 + 6 4 = 1 0 0
The cubes can be found by adding the numbers "around the corner", i.e.
1
=
1
2
+
4
+
2
=
8
3
+
6
+
9
+
6
+
3
=
2
7
4
+
8
+
1
2
+
1
6
+
1
2
+
8
+
4
=
6
4
so that everything is accounted for.
That's a good answer. I have found this equation when I was 13 years old .And I haven't the answer until I am 16 years old . You are excellent.
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Well, this isn't actually a "proof". Rather, it's an interesting observation that can be made about the multiplication table. Actually proving this takes a few more steps.
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can you show ?
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@Nguyễn Hưng – Sure. For this example where n = 4
First note that
1
+
2
+
3
+
4
+
2
+
4
+
6
+
8
+
3
+
6
+
9
+
1
2
+
4
+
8
+
1
2
+
1
6
=
1
(
1
+
2
+
3
+
4
)
+
2
(
1
+
2
+
3
+
4
)
+
3
(
1
+
2
+
3
+
4
)
+
4
(
1
+
2
+
3
+
4
)
=
( 1 + 2 + 3 + 4 ) 2
Then note that
4
+
8
+
1
2
+
1
6
+
1
2
+
8
+
4
=
(
4
+
1
2
)
+
(
8
+
8
)
+
(
1
2
+
4
)
+
(
1
6
)
=
1
6
+
1
6
+
1
6
+
1
6
=
4
(
1
6
)
=
4
3
which is the same pattern for other cubes, so that we end up with
( 1 + 2 + 3 + 4 ) 2 = 1 3 + 2 3 + 3 3 + 4 3
In vietnam ,the children don't study by multiplication table similar to your country .
I read the question wrong... The answer can also be demonstrated using the formula for sums.
yes .You are true
1 3 + 2 3 + ⋯ + n 3 = 4 n 2 ( n + 1 ) 2 = 2 n ( n + 1 ) = 1 + 2 + ⋯ + n