Question

Tính

\(S^4=1^2+2^2+3^2+...+49^2+50^2\)

\(S^5=1^3+2^3+3^3+...+49^3+50^3\)

Answer 1

S⁴ = 1² + 2² + 3² + ... + 49² + 50²

S⁴ = 1 + 2 ( 1 + 1 ) + 3 ( 2 + 1 ) + ... + 49 ( 48 + 1 ) + 50 ( 49 + 1 )

S⁴ = 1 + 1.2 + 2 + 2.3 + 3 + ... + 48 . 49 + 49 + 49 . 50 + 50

S⁴ = ( 1 + 2 + 3 + ... 49 + 50 ) + ( 1.2 + 2.3  + ... + 48 . 49 + 49 . 50 )

đặt A = 1 + 2 + 3 + ... 49 + 50

Ta tính được : A = 1275

đặt B = 1.2 + 2.3  + ... + 48 . 49 + 49 . 50

3B = 1.2.3 + 2.3.3 + ... + 48.49.3 + 49.50.3

3B = 1.2.3 + 2.3.(4-1) + ... + 48.49.(50-47) + 49.50.(51-48)

3B = 1.2.3 + 2.3.4 - 1.2.3 + ... + 48.49.50 - 47.48.49 + 49.50.51-48.49.50

3B = 49.50.51

B = 49.50.51 : 3 =  41650

=> S⁴ = 41650 + 1275 = 42925

S⁵ = 1³ + 2³ + 3³ + ... 49³ + 50³

S⁵ = 1 + 2² ( 1 + 1 ) + 3² ( 2 + 1 ) + ... 49² ( 48 + 1 ) + 50² ( 49 + 1 )

S⁵ = 1² + 1.2² + 2² + 2.3² + 3² + ... + 48.49² + 49² + 49.50² + 50²

S⁵ = ( 1² + 2² + 3² + ... + 49² + 50² ) + ( 1.2² + 2.3² + ... + 48.49² + 49.50² )

đặt Y = 1² + 2² + 3² + ... + 49² + 50² 

Y = 42925

đặt M = 1.2² + 2.3² + ... + 48.49² + 49.50² 

M = 1.2.(3-1) + 2.3.(4-1) + ... + 48.49.(50-1) + 49.50.(51-48)

M = (1.2.3+2.3.4+...+48.49.50+49.50.51)-(1.2+2.3+...+48.49+49.50)

đến đây đơn giản rồi

Answer 2

 

Tính

S4=12+22+32+...+492+502S^4=1^2+2^2+3^2+...+49^2+50^2S4=12+22+32+...+492+502

S5=13+23+33+...+493+503S^5=1^3+2^3+3^3+...+49^3+50^3S5=13+23+33+...+493+503