Sum of the Series 1-2 3-4 5-6 to n terms

The given series, 1-2 3-4 5-6, consists of pairs of consecutive numbers with alternating signs. Each pair consists of a positive number followed by its negative counterpart. To find the sum of this series up to n terms, we need to understand the pattern and apply the appropriate formula.

Understanding the Pattern

Let's examine the first few terms of the series:

1-2 3-4 5-6 7-8 9-10 ...

From the pattern, it can be observed that the positive numbers are the odd terms (1, 3, 5, 7, ...) and the negative numbers are the even terms (2, 4, 6, 8, ...).

Formulating the Series

We can express the series as follows:

1 - 2 + 3 - 4 + 5 - 6 + 7 - 8 + 9 - 10 + ...

Let's denote the sum of the series as S. We can write the sum of the positive terms as S+ and the sum of the negative terms as S-.

Sum of Positive Terms (S+)

The positive terms in the series are the odd terms. The sum of the odd numbers up to n terms can be calculated using the formula:

S+ = (n/2)^2

This formula represents the sum of an arithmetic series with a common difference of 2.

Sum of Negative Terms (S-)

The negative terms in the series are the even terms. The sum of the even numbers up to n terms can be calculated using the formula:

S- = -((n/2)(n/2 + 1))

This formula represents the sum of an arithmetic series with a common difference of 2, multiplied by -1.

Sum of the Series (S)

The sum of the series can be calculated by subtracting the sum of the negative terms from the sum of the positive terms:

S = S+ - S-

Example Calculation

Let's calculate the sum of the series up to 4 terms:

Positive terms (S+):
S+ = (4/2)^2 = 2^2 = 4

Negative terms (S-):
S- = -((4/2)(4/2 + 1)) = -((2)(3)) = -6

Sum of the series (S):
S = S+ - S- = 4 - (-6) = 4 + 6 = 10

Therefore, the sum of the series 1-2 3-4 5-6 up to 4 terms is 10.

Conclusion

To find the sum of the series 1-2 3-4 5-6 to n terms, we can use the formulas for the sum of positive and negative arithmetic series. By subtracting the sum of the negative terms from the sum of the positive terms, we can obtain the sum of the entire series.