SOLUTION: Assume 1^2 + 2^2 + 3^2 +... n^2 = (n(n+1)(2n+1))/6 is true for all positive integers n. If we replace the right hand side with ((n+2)(n+3)(2n+5))/6 , what term(s) do we add to th

Question 972088: Assume 1^2 + 2^2 + 3^2 +... n^2 = (n(n+1)(2n+1))/6 is true for all positive integers n. If we replace the right hand side with ((n+2)(n+3)(2n+5))/6 , what term(s) do we add to the left hand side?
I believe the correct answer is (n+1)^2 but I want to double check
Answer by Edwin McCravy(20056) (Show Source): You can put this solution on YOUR website!
Assume 1^2 + 2^2 + 3^2 +... n^2 = (n(n+1)(2n+1))/6 is true for all positive integers n. If we replace the right hand side with ((n+2)(n+3)(2n+5))/6 , what term(s) do we add to the left hand side?
I believe the correct answer is (n+1)^2 but I want to double check

-------------------------------------------------------------

I'm afraid that's not all but only part of what is added to the left side.

---------------------------------------------------------------



To find out what we have added to the left hand side when we have replaced
the right side with  we compare them:

  

We see that the first factor on the left numerator is n, while the first
factor on the right numerator is (n+2), so that makes us suspect that n 
has been replaced by (n+2).  

We check to see if that is the case with the second factor on the left. We take
(n+1) and replace n by n+2 in it and we get (n+2+1) or (n+3) which is the second
factor on the right.  So far so good.

We only need to show that if we replace n by n+2 in the third factor on the left
that we will get the third factor on the right (2n+5).  We show that by 
replacing n by n+2 in (2n+1):

(2(n+2)+1) = (2n+4+1) = (2n+5)

Now we know that the right side has been replaced by (n+2) throughout.
Therefore we know that the sum on the left, which is the sum up through n terms,
is now to be carried up to (n+2) terms, which is 2 more terms, so now we
have

 

Therefore two more terms  have been added to the left,
not just the one term that you were thinking was added.

Edwin

RELATED QUESTIONS
I am trying to find the steps to solve this one. Any help will be appreciated! Thanks! (answered by robertb)

Dear math teacher, I am having difficulties with the following problem: 4 times nC2 (answered by Theo)

show that 1(1!)+2(2!)+3(3!)+...+n(n!)= (n+1)!-1 for all positive integers... (answered by venugopalramana)

Prove that for all positive integers n,... (answered by Edwin McCravy)

n^3-2n^2(n+1) (answered by MathLover1)

use mathematical induction to prove that 1^2 + 2^2 + 3^2 +...+ n^2 = n(n+1)(2n+1)/6... (answered by solver91311)

Find all integers n for which \frac{n^2 + n + 1}{n - 2 + n^3} is an integer. (answered by CPhill,math_tutor2020)

I've been working through factoring polynomials and have solved the following as such;... (answered by stanbon)

I need help with this mathematical induction to show that the given statement is true for (answered by jim_thompson5910)