What is the Sum of the Alternating Sequence 1, -1, 3, 1, 3, 5, …, to N Terms?

When dealing with sequences and their sums, it’s fascinating to explore different patterns and series. One such intriguing series is the alternating sequence consisting of odd numbers, where the signs alternate between positive and negative. The sequence starts as 1, -1, 3, 1, 3, 5, and so on. This article will delve into the pattern of the sequence and the sum to n terms, providing a detailed explanation and a step-by-step guide for finding the sum.

Understanding the Sequence

The sequence we are examining is: 1, -1, 3, 1, 3, 5, … and so forth. Each term after the first can be described in terms of the position of the term in the sequence. The term can be expressed as:

For odd terms, the term is positive and given by the position in the sequence (i.e., the term is 1, 3, 5, 7, etc.). For even terms, the term is negative and given by the position in the sequence (i.e., the term is -1, 1, -3, 3, etc.).

Formulating the Sum of N Terms

Let’s denote the sum of the first n terms of the sequence by Sn. We need to find a general formula for this sum.

Sum of Positive Odd Terms

The positive odd terms in the sequence can be listed as:

1 3 5 7 … 2n-1

This is an arithmetic progression (AP) with the first term A 1, common difference D 2, and the last term L 2n-1. The number of terms in this AP is n.

The sum of the terms in an AP can be calculated using the formula:

Tn n/2 (A L)

In our case:

Tn n/2 (1 (2n-1)) n/2 (2n) n2

Sum of Negative Odd Terms

The negative odd terms in the sequence can be listed as:

-1 1 -3 3 … -(2n-1)

This is also an arithmetic progression with the first term A -1, common difference D 2, and the last term L -(2n-1). The number of terms in this AP is n.

The sum of the terms in an AP can be calculated using the formula:

Tn n/2 (A L)

In our case:

Tn n/2 (-1 - (2n-1)) n/2 (-2n) -n2

Total Sum of the Sequence

The total sum Sn of the sequence is the sum of the positive and negative terms:

Sn n2 (-n2) 0

This is only true if the number of positive and negative terms is equal, i.e., n is odd.

For an even number of terms, the sum of the positive terms and the sum of the negative terms can be added directly without cancellation:

Sn (1 - 1) (3 1) (5 - 3) … (2n-1 (n-1))

Each pair sums to 2, and there are n/2 pairs, so:

Sn 2 * (n/2) n

General Formula for the Sum of Squares

Notice that the terms in the sequence involve squares of integers. The general sum of the squares of the first n odd numbers can be written as:

Sn 12 (-1)2 32 12 32 52 … (2n-1)2

Using the formula for the sum of the squares of the first n natural numbers:

Sn (12 22 32 … n2) n(n 1)(2n 1)/6

Since the sequence alternates between positive and negative squares, half of the squares are positive and the other half are negative. Therefore, the net sum is:

Sn n(n 1)(2n 1)/6

For the specific sequence given, the sum of the squares to n terms is:

Sn n(n 1)(2n 1)/6

Conclusion

The sum of the alternating sequence of odd numbers (1, -1, 3, 1, 3, 5, …) to n terms can be determined using the alternating arithmetic progression and the sum of the squares of the first n terms. With an understanding of the sequence’s pattern and the application of appropriate formulas, the sum can be calculated accurately.

Keywords

sum of sequence, alternating sequence, sum of squares