Question

Find the given sum:
1.2.3 + 2.3.4 +3.4.5 + ... + n(n+1)(n+2)

Answer 1

4.5.6 +5.6.7+6.7.8+7.8.9+

Answer 2

Answer:

n(n+1)(n+2)(n+3)/4

Step-by-step explanation:

1.2.3 + 2.3.4 +3.4.5 + ... + n(n+1)(n+2)

= ∑ n(n+1)(n+2)

= ∑ n(n² + 3n + 2)

= ∑ n³ + 3n²  + 2n

=  n²(n+1)²/4  + 3(n)(n+1)(2n+1)/6 + 2n(n+1)/2

= n²(n+1)²/4  + (n)(n+1)(2n+1)/2 + n(n+1)

= (n(n+1)/2 )( n(n+1)/2  + (2n+1) + 2)

= (n(n+1)/4 )( n(n+1)  + 2(2n+1) + 4)

= (n(n+1)/4 )( n² + n + 4n+2 + 4)

= (n(n+1)/4 )( n² + 5n + 6)

= n(n+1)( n² + 5n + 8)/4

= n(n+1)(n+2)(n+3)/4

1.2.3 + 2.3.4 +3.4.5 + ... + n(n+1)(n+2)  = n(n+1)(n+2)(n+3)/4