n∑k=1,(2k^2-4k+3)=n/3(2n^2-3n+4),prove by mathematical induction.

Question

I stucked with this after step one where u prove by substituting 1 into the unknowns.Do kindly assist.

Answer 1

That's an important first step.  You've proven that it works for n = 1.  In order to use induction, we need to prove that if it's true for n-1, then it must be true for n.  Let S_n be the sum through n.  Then,

 

S_n = S_n-1 + 2n² - 4n + 3    (based on the definition of our sum)

 

Sn = (n-1)/3 * (2(n-1)² - 3(n-1) + 4)   +    2n² - 4n + 3      (based on substituting in our formula for S_n-1)

 

Now, all you need to do is multiply out and simplify this expression and demonstrate that it comes out to 

n/3(2n² - 3n + 4)

 

If you need additional help with that, feel free to let me know.