Question

Prove that 1square + 2square + 3square .......+n square =n(n+1)2n+1 divided by 6​

Answer 1

Answer:

Step-by-step explanation:

We have that: 

1² + 2² + 3² + ... + n² = n(n + 1)(2n + 1)/6 

We can prove this by induction. First, we need to show that it is true for n = 1. 

1(1 + 1)(2 + 1)/6 = 1 

So it is true for n = 1. 

Now, we need to show that if it is true for a value, then it is true for the next value (which makes it true for all integers greater than that value as well). 

Adding (n + 1)² to both sides gives us: 

1² + 2² + 3² + ... + n² + (n + 1)² = n(n + 1)(2n + 1)/6 + (n + 1)² 

Focusing on the RHS: 

n(n + 1)(2n + 1)/6 + (n + 1)² 

= n(n + 1)(2n + 1)/6 + 6(n + 1)²/6 

= [n(n + 1)(2n + 1) + 6(n + 1)²]/6 

= (n + 1)[n(2n + 1) + 6(n + 1)]/6. . . . . . . . . .Factor out (n + 1) 

= (n + 1)(2n² + n + 6n + 6)/6 

= (n + 1)(2n² + 4n + 3n + 6)/6 

= (n + 1)[2n(n + 2) + 3(n + 2)]/6 

= (n + 1)(2n + 3)(n + 2)/6 

= (n + 1)[2(n + 1) + 1][(n + 1) + 1]/6 

Since this result is the same as before except n + 1 is in n's spot. It is true for all n equal to or greater than 1! Q.E.D. 

Answers

1^2 + 2^2 + 3^2 + ... + n^2 = n(n+1)(2n+1)/6

Answer 2

Answer:

hope you UNDERSTOOD this