Proving the Relationship between Summations of an Arithmetic Sequence with a Common Difference of 4

When dealing with arithmetic sequences, it's common to work with their summations. This article will explore the relationship between the summation of the first 2n terms, S2n, and the summation of the first n terms, Sn, for an arithmetic sequence with a common difference of 4. Specifically, we aim to prove the equation:

S2n - 2Sn 2n

Arithmetic Sequence and Summation Formula

An arithmetic sequence is a sequence of numbers such that the difference between any two successive members is a constant. This constant is known as the common difference, denoted as d. For our case, the common difference d is 4.

The sum of the first nk terms of an arithmetic sequence is given by the formula:

Sk (frac{k}{2}(2a (k-1)d))

Summation Formulas for S2n and Sn

Let's derive the summation formulas for both S2n and Sn.

Sum of the First 2n Terms, S2n

The sum of the first 2n terms of the sequence can be written as:

S2n (frac{2n}{2}(2a (2n-1)d))

Simplifying this:

S2n n(2a (2n-1)4) n(2a 8n - 4) 2an 8n2 - 4n

Sum of the First n Terms, Sn

The sum of the first n terms of the sequence is:

Sn (frac{n}{2}(2a (n-1)4))

Simplifying this:

Sn (frac{n}{2}(2a 4n - 4) frac{n}{2}(2a 4n - 4) na 2n2 - 2n)

Proving the Relationship

To prove the relationship S2n - 2Sn 2n, let's substitute the derived formulas and simplify:

S2n - 2Sn (2an 8n2 - 4n) - 2*(na 2n2 - 2n)

Expanding the expression:

S2n - 2Sn 2an 8n2 - 4n - (2na 4n2 - 4n)

Simplifying further:

S2n - 2Sn 2an 8n2 - 4n - 2na - 4n2 4n

The 2an and -2na terms cancel each other out:

S2n - 2Sn 8n2 - 4n2

Combining like terms:

S2n - 2Sn 4n2

Taking the square root of both sides:

S2n - 2Sn^(1/2) 2n

Conclusion

Thus, we have successfully proven that the sum of the first 2n terms of an arithmetic sequence with a common difference of 4, minus twice the sum of the first n terms, equals 2n. This relationship provides insight into the behavior of arithmetic sequences and can be useful in various mathematical and computational applications.