Question

Prove
that
Sn=Σn=[n(n-1)]/2
​

Answer 1

Answer:

thia can be solved by using the DYS formula that is do yourself

Answer 2

Answer:

In this problem, we're required to prove that...

Eq1.png

Or, written explicitly...

Eq2.png for all natural numbers (i.e. n = 1, 2, 3, ...)

Since this is an ordered set, we can use Mathematical Induction to prove this statement to be true. There really are only 2 major steps when in comes to the inductive method...

Step 1: The Basis Test

This is to see if the equation holds true for first number in the set. In this case, n = 1. Thus, on the left-hand side (LHS) we have:

Eq3.png

On the right-hand side (RHS) we have:

Eq4.png

So this equation satisfies the basis test, in that when n = 1, LHS = RHS = 2. So we're satisfied that the equation holds true so far.

Step 2: The Inductive Test

This is to establish that if the equation holds true for any value in the order (i.e. n = k, where k = 1, 2, 3,...), then the equation holds true for the next value in the order (i.e. n = k +1).

The most convenient (and according to some, the best) way to approach this is to assume that the equation holds true for n = k. So, in our case, this means...

Eq5.png

Then, if this assumption is correct, we need to prove that the above holds true for n = k +1. Therefore we need to prove the following...

Eq6.png

Now, consider the LHS...

Eq7.png

Up to the k(k + 1) term, we have...

Eq9.png

...since we assumed the equation holds true for when n = k.

Now...

Eq10.png

And thus we have proved: Σn(n + 1) = n(n + 1)(n + 2)/3 for all natural numbers by mathematical induction.

Please view the video below if the explanation above was not clear.