Prove that 1square + 2square + 3square .......+n square =n(n+1)2n+1 divided by 6
Answers
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:
hope you UNDERSTOOD this