Use mathematical induction to prove the statement is true for all positive integers n, or show why it is false. 1^2 + 4^2 + 7^2 + + (3n - 2)^2 = [n(6n^2-3n-1)/2]

We start by proving the statement is true for . In this case, the left hand side is simply:

The right hand side is:

So the property holds for .

Now we assume the property is valid for some and prove that it implies that it is also valid for . That means we shall obtain:

We start by writing the property for and recognize that we've only added an extra term:

We now expand the square and merge everything together in a single fraction:

Since the desired expression has a facto, we may now divide the numerator by , using, for instance, Ruffini's rule, in order to find it:

So we get:

We now add and subtract and in order to obtain the form corresponding to , since we know that it must appear:

So we finally got:

which is the desired result. So we proved that the statement is true for every integer .

1/4 and 8/3 because:

the reciprocal if 4 is 1/4 because 4 is used as 4/1 and the opposite of that is 1/4. the opposite of 3/8 is 8/3 so that is the answer.

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step-by-step explanation: