A = 1 × 2 × 3 + 2 × 3 × 4 + .....+ 48 × 49 × 50

ta có 4 x A = 1 x 2 x 3 x 4 + 2 x 3 x 4 x (5 -1) + .....+ 48 × 49 × 50 x (51 - 47)

= 1 x 2 x 3 x 4 +  2 x 3 x 4 x 5 - 1 x 2 x 3 x 4 + ... + 48 x 49 x 50 x 51 - 47 x 48 x 49 x 50

= 48 x 49 x 50 x 51

suy ra A = (48 x 49 x 50 x 51) : 4

              = 12 x 49 x 50 x 51

nhớ k cho mik nha rùi mik lm nốt cho

A = 1 × 2 × 3 + 2 × 3 × 4 + .....+ 48 × 49 × 50

ta có 4 x A = 1 x 2 x 3 x 4 + 2 x 3 x 4 x (5 -1) + .....+ 48 × 49 × 50 x (51 - 47)

= 1 x 2 x 3 x 4 +  2 x 3 x 4 x 5 - 1 x 2 x 3 x 4 + ... + 48 x 49 x 50 x 51 - 47 x 48 x 49 x 50

= 48 x 49 x 50 x 51

suy ra A = (48 x 49 x 50 x 51) : 4

              = 12 x 49 x 50 x 51

1/ S=1.2+2.3+3.4+...+50.51

=> 3S=1.2.3+2.3.3+3.4.3+...+50.51.3

=> 3S=1.2.3+2.3.(4-1)+3.4.(5-2)+...+50.51(52-49)

=> 3S=(1.2.3+2.3.4+3.4.5+...+50.51.52)-(1.2.3+2.3.4+...+49.50.51)

=> 3S=50.51.52 => S=50.51.52:3=44200

Đáp số: 44200

2/ A=1²+2²+3²+4²+...+50² = 1(2-1)+2(3-1)+3(4-1)+...+50(51-1)

=> A=(1.2+2.3+3.4+...+50.51)-(1+2+3+...+50)

=> A=S-\(\frac{50\left(50+1\right)}{2}\)=44200-1275

A=42925

Đáp số: 42925

a, Ta có : S = 1*2 + 2*3 +3*4 + .... + 50*51

3S=1*2*3+2*3*3+3*4*3+....+50*51*3

3S=1*2*3+2*3*(4-1)+3*4*(5-2)+....+50*51*(52-49)

3S=1*2*3+2*3*4-1*2*3+3*4*5-2*3*4+...+50*51*52-49*50*51

3S=50*51*52

S=(50*51*52)/3=442000

b,Ta có   1² + 2² + 3² + ....... + n²=\(\frac{n\cdot\left(n+1\right)\cdot\left(2n+1\right)}{6}\)

=>   1² + 2² + 3² + ....... + 50²= \(\frac{50\cdot\left(50+1\right)\cdot\left(2\cdot50+1\right)}{6}\)

=\(\frac{50\cdot51\cdot101}{6}\)= 42925

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

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

\(=\left(1-\frac{1}{3^2}\right)\left(1-\frac{1}{4^2}\right)\left(1-\frac{1}{5^2}\right)...\left(1-\frac{1}{50^2}\right)\)

\(=\frac{8}{3\cdot3}\cdot\frac{15}{4\cdot4}\cdot\frac{24}{5\cdot5}\cdot....\cdot\frac{2499}{50\cdot50}\)

\(=\frac{\left(2\cdot4\right)\left(3\cdot5\right)\left(4\cdot6\right)...\left(49\cdot51\right)}{\left(3\cdot3\right)\left(4\cdot4\right)\left(5\cdot5\right)...\left(50\cdot50\right)}\)

\(=\frac{\left(2\cdot3\cdot4\cdot...\cdot49\right)\left(4\cdot5\cdot6\cdot...\cdot51\right)}{\left(3\cdot4\cdot5\cdot...\cdot50\right)\left(3\cdot4\cdot5\cdot...\cdot50\right)}\)

\(=\frac{2\cdot51}{50\cdot3}\)