# Prove the statement by mathematical induction?

##
**If **#S_n=1+2xx2^2+3^2+2xx4^2+5^2+2xx6^2# **then ** #S_n=n(n+1)^2/2,n # #is # even

#S_n=n^2(n+1)/2,n# #is # odd

**If #S_n=1+2xx2^2+3^2+2xx4^2+5^2+2xx6^2# **

**then**

##### 1 Answer

#### Answer:

See below.

#### Explanation:

Since we have two different situations, let's break down this into two different statements,

if

if

Let's prove

Now, let

We must prove if the next case holds. However, the next case is **not** **odd** positive integers. This means that the next case is actually

Notice how the part highlighted in red is the former sum,

We know that

So by substituting this into the statement, we get

This is fairly easy to find. Multiply both sides by

Which is true. Thus, by mathematical induction,

The second statement,

If you do all that, you will find that

Hence, by **Mathematical induction**, the statement

is true for all