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Conta-se que um professor de Matemática mandou aos alunos de sua turma que somassem de 1 a 100 como forma de castigo. O professor ficou surpreso quando um dos alunos, Carl Friedrich Gauss , havia feito a soma corretamente em pouco tempo de uma outra forma.
Eis a ideia de Gauss:
Soma de 1 a 100:
S
=
1
+
2
+
3
+
⋯
+
99
+
100
{\displaystyle S=1+2+3+\dots +99+100}
A ordem não altera a soma:
S
=
100
+
99
+
98
+
⋯
+
2
+
1
{\displaystyle S=100+99+98+\dots +2+1}
Soma-se de membro a membro as duas igualdades:
S
=
1
+
2
+
3
+
⋯
+
99
+
100
{\displaystyle S=1+2+3+\dots +99+100}
S
=
100
+
99
+
98
+
⋯
+
2
+
1
{\displaystyle S=100+99+98+\dots +2+1}
______________________
S
+
S
=
(
1
+
100
)
+
(
2
+
99
)
+
(
3
+
98
)
+
⋯
+
(
99
+
2
)
+
(
100
+
1
)
{\displaystyle S+S=(1+100)+(2+99)+(3+98)+\dots +(99+2)+(100+1)}
2
S
=
101
+
101
+
101
+
101
+
⋯
+
101
{\displaystyle 2S=101+101+101+101+\dots +101}
2
S
=
101
∗
100
{\displaystyle 2S=101*100}
S
=
[
100
(
100
+
1
)
]
/
2
{\displaystyle S=[100(100+1)]/2}
S
=
5050
{\displaystyle S=5050}
Generalizando:
S
=
1
+
2
+
3
+
⋯
+
(
n
−
2
)
+
(
n
−
1
)
+
n
{\displaystyle S=1+2+3+\dots +(n-2)+(n-1)+n}
2
S
=
(
n
+
1
)
+
(
n
+
1
)
+
(
n
+
1
)
+
⋯
+
(
n
+
1
)
{\displaystyle 2S=(n+1)+(n+1)+(n+1)+\dots +(n+1)}
2
S
=
n
(
n
+
1
)
{\displaystyle 2S=n(n+1)}
S
=
[
n
(
n
+
1
)
]
/
2
{\displaystyle S=[n(n+1)]/2}
Logo,
1
+
2
+
3
+
⋯
+
n
=
[
n
(
n
+
1
)
]
/
2
{\displaystyle 1+2+3+\dots +n=[n(n+1)]/2}
1)
S
=
1
+
2
+
3
+
⋯
+
68
+
69
+
70
{\displaystyle S=1+2+3+\dots +68+69+70}
S
=
[
70
(
70
+
1
)
]
/
2
{\displaystyle S=[70(70+1)]/2}
S
=
(
70
∗
71
)
/
2
{\displaystyle S=(70*71)/2}
S
=
2485
{\displaystyle S=2485}
2)
M
=
41
+
42
+
43
+
⋯
+
150
{\displaystyle M=41+42+43+\dots +150}
Neste caso existem duas formas de encontrar a resposta.
1ª forma :
M
=
41
+
42
+
43
+
⋯
+
150
{\displaystyle M=41+42+43+\dots +150}
M
=
(
40
+
1
)
+
(
40
+
2
)
+
(
40
+
3
)
+
⋯
+
(
40
+
110
)
{\displaystyle M=(40+1)+(40+2)+(40+3)+\dots +(40+110)}
M
=
(
40
+
40
+
40
+
⋯
+
40
)
+
(
1
+
2
+
3
+
⋯
+
110
)
{\displaystyle M=(40+40+40+\dots +40)+(1+2+3+\dots +110)}
M
=
(
40
∗
110
)
+
[
(
110
∗
111
)
/
2
]
{\displaystyle M=(40*110)+[(110*111)/2]}
M
=
4400
+
6105
{\displaystyle M=4400+6105}
M
=
10505
{\displaystyle M=10505}
2ª forma:
M
=
41
+
42
+
43
+
⋯
+
150
{\displaystyle M=41+42+43+\dots +150}
M
=
(
1
+
2
+
3
+
⋯
+
150
)
−
(
1
+
2
+
3
+
⋯
+
40
)
{\displaystyle M=(1+2+3+\dots +150)-(1+2+3+\dots +40)}
M
=
[
(
150
∗
151
)
/
2
]
−
[
(
40
∗
41
)
/
2
]
{\displaystyle M=[(150*151)/2]-[(40*41)/2]}
M
=
11325
−
820
{\displaystyle M=11325-820}
M
=
10505
{\displaystyle M=10505}
![{\displaystyle S=[100(100+1)]/2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dd174c1bd79fcfbee8d71797f8a79dbeb90b479a)
![{\displaystyle S=[n(n+1)]/2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ec33cdb132f60ab092b13516552903556bb00afc)
![{\displaystyle 1+2+3+\dots +n=[n(n+1)]/2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5a08ebe94614544ce1d2106f2c963ebc9d629064)
![{\displaystyle S=[70(70+1)]/2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cfaa3d53e3ed5b1fec36e18951b9b51e61ca965b)
![{\displaystyle M=(40*110)+[(110*111)/2]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0fe9fbe305109a003756b2e9f0573f769a3d8c4c)
![{\displaystyle M=[(150*151)/2]-[(40*41)/2]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bfc2a828eeafd20f8eaaee59d1beaf242301a98e)
