one plus two is three. one plus two plus three is six. one plus two plus three plus four is ten. if we keep doing this long enough we'll find that the sum of numbers from 1 to n is a perfect square. find the three first numbers.
2006-12-02 00:46:00 · 9 answers · asked by Anonymous in Science & Mathematics ➔ Mathematics
No more calculators! Let's do this theoretically.
Recall that the sum of the numbers from 1 to n
is called the nth triangular number, t_n and that
if we add the first n positive integers, we get t_n = n(n+1)/2.
(If you're not familiar with these check
Wikipedia for more info.)
So what you are really asking is: Which triangular
numbers are squares? I'll derive a formula
for them.
Suppose m² = n(n+1)/2 = (n² + n)/2
Then 2m² = n² + n.
The idea now is to make the right-hand side
a square. To do this, multiply both sides by 4
and add 1.
We get
8m² + 1 = (2n+1)².
Let x = 2n + 1.
Then x² - 8m² = 1.
Now we have a Pell equation to solve.
The theory of this type of Diophantine
equation tells us that if (x, m) is
the smallest (or fundamental) solution,
then all other solutions are obtained from
x_k + m_k*sqrt 8 = (x + m*sqrt 8)^k.
For our problem, the fundamental solution is
(x, m) = (3,1)
So all positive integer solutions are given by
x_k + m_k*sqrt 8= (3 + sqrt 8)^k.
So let's find the first 4 solutions by setting k = 1, 2, 3, and 4.
x m n
3 1 1
17 6 8
99 35 49
577 204 288
and you can readily check that n(n+1)/2 is equal to
m² in each case.
2006-12-02 02:28:32 · answer #1 · answered by steiner1745 7 · 0⤊ 0⤋
The person who asked you this question left out one word: "odd"
1+3 = 4 - a perfect square
1+3+5 = 4+5 = 9 - a perfect square
9+7 = 16 - a perfect square.
Now, why is this?
let's look at the difference between perfect squares.
Say n^2 and (n+1)^2
(N+1)*(N+1) = N*N + 2*N+1
The difference is
N*N + 2*N + 1 - N*N = 2*N + 1 which is equal to (N) + (N+1)
N+N+1 is ALWAYS odd, interestingly, and when you increment N by 1, you increment N+N+1 by 2
So IF the difference between two consecutive squares is, say, 14565 (an odd number), then the difference between the next two consecutive squares is 14567 and you don't have to do anything fancy to figure that out!
2006-12-02 08:55:02 · answer #2 · answered by firefly 6 · 0⤊ 0⤋
I'm answering your question as is, without modifications.
if you continue adding numbers, you'll eventually get numbers which happen to be perfect squares.
When adding numbers in this way, you get the following pattern:
1
3
6
10
15
21
28
36 <-- first perfect square (of 6)
45
55
66
78
91
105
120
136
153
171
190
210
231
253
276
300
325
351
378
406
435
465
496
528
561
595
630
666
703
741
780
820
861
903
946
990
1035
1081
1128
1176
1225 (<-- second perfect square (of 35))
1275
1326
1378
1431
1485
1540
1596
1653
1711
1770
1830
1891
1953
2016
2080
2145
2211
2278
2346
2415
2485
2556
2628
2701
2775
2850
2926
3003
3081
3160
3240
3321
3403
3486
3570
3655
3741
3828
3916
4005
4095
4186
4278
4371
4465
4560
4656
4753
4851
4950
5050
5151
5253
5356
5460
5565
5671
5778
5886
5995
6105
6216
6328
6441
6555
6670
6786
6903
7021
7140
7260
7381
7503
7626
7750
7875
8001
8128
8256
8385
8515
8646
8778
8911
9045
9180
9316
9453
9591
9730
9870
10011
10153
10296
10440
10585
10731
10878
11026
11175
11325
11476
11628
11781
11935
12090
12246
12403
12561
12720
12880
13041
13203
13366
13530
13695
13861
14028
14196
14365
14535
14706
14878
15051
15225
15400
15576
15753
15931
16110
16290
16471
16653
16836
17020
17205
17391
17578
17766
17955
18145
18336
18528
18721
18915
19110
19306
19503
19701
19900
20100
20301
20503
20706
20910
21115
21321
21528
21736
21945
22155
22366
22578
22791
23005
23220
23436
23653
23871
24090
24310
24531
24753
24976
25200
25425
25651
25878
26106
26335
26565
26796
27028
27261
27495
27730
27966
28203
28441
28680
28920
29161
29403
29646
29890
30135
30381
30628
30876
31125
31375
31626
31878
32131
32385
32640
32896
33153
33411
33670
33930
34191
34453
34716
34980
35245
35511
35778
36046
36315
36585
36856
37128
37401
37675
37950
38226
38503
38781
39060
39340
39621
39903
40186
40470
40755
41041
41328
41616 (<-- third perfect square (of 204))
I just used an excel spreadsheet to do the math (Column A was 1, 2, 3, 4....; column B was adding them up (=A2+B1) and column C was square root of column B.
2006-12-02 09:23:07 · answer #3 · answered by sep_n 3 · 1⤊ 0⤋
If n is 1, the sum is 1 which is a perfect square.
If n = 8, the sum is 36 which is a perfect square.
If n = 49 the sum is 1225 which is a perfect square. (The next one after that is 288 which sums to 41616.)
2006-12-02 09:24:08 · answer #4 · answered by hayharbr 7 · 1⤊ 0⤋
Here is the first 8 numbers:
1
8
49
288
1681
3074
5044
6437
Try to find another number by yourself
2006-12-02 09:39:20 · answer #5 · answered by iyiogrenci 6 · 0⤊ 0⤋
the sum is a square if you add odd numbers so forget about the even ones =)
2006-12-02 08:52:59 · answer #6 · answered by Anonymous · 0⤊ 1⤋
-1, 2, 3
2006-12-02 08:49:48 · answer #7 · answered by Purple Tears 1 · 0⤊ 3⤋
I'm sure glad I'm not is school anymore! I have no idea. Good Luck to you!
2006-12-02 08:54:36 · answer #8 · answered by Shari 5 · 0⤊ 1⤋
no idea
2006-12-02 08:54:04 · answer #9 · answered by Anonymous · 0⤊ 1⤋