The first three terms of an arithmetic series are 6p+2, 4p²-10 and 4p+3 respectively. Find the possible values of p. Calculate the common difference for each value of p.

First Case:

Second Case:

Step-by-step explanation:

We know that the first three terms of an arithmetic series are:

Since this is an arithmetic sequence, each subsequent term is d more than the previous term, where d is our common difference.

Therefore, we can write the second term as;

And, likewise, for the third term:

Let's solve for d for each of the equations.

Subtracting in the first equation yields:

And for the second equation:

To avoid fractions, let's multiply the first equation by 2. Hence:

Therefore:

Simplifying yields:

Solve for p. We can factor:

Factor:

Grouping:

Zero Product Property:

Then, we can use the second equation to solve for d. So:

Substituting:

So, for the first case, p is 5/2 and d is -2.

Likewise, for the second case:

So, for the second case, p is -5/4, and d is 7/4.

By using the values, we can determine our series.

For Case 1, we will have:

17, 15, 13.

For Case 2, we will have:

-11/2, -15/4, -2.

(n^3 - n)

= n(n-1)(n+1)

since it is divisible by 3 and 2 ,because it is a multiple 3 consecutive number

if, n-1 is odd then n+1 is also odd, then it is a multiple of 9

but if, n-1 is even then n+1 is also even, then it is a multiple of 4

so, largest multiple = max(9*2, 4*3) = max(18, 12) = 18

in mathematics, a function is a relationship between two variables such that the value of one of the variables, called the dependent variable, is determined by the value of the other variable, called the independent variable. there is a specific term we use to describe the set of independent variables of a function.