Question

The sum to \(n\) terms of the series, where \(n\) is an even number : \(1^{2}-2^{2}+3^{2}-4^{2}+5^{2}-6^{2}+\ldots:\) (a) \(n(n+1)\) (b) \(\frac{n(n+1)}{2}\) (c) \(-\frac{n(n+1)}{2}\) (d) none of these

Short answer

Answer: (c) \(-\frac{n(n+1)}{2}\)

Step-by-step solution

Identify the even and odd terms in the series

The series can be represented in two parts, one representing the sum of the squares of the positive terms, and the other representing the sum of the squares of the negative terms: Even terms: \(-2^{2}, -4^{2}, -6^{2}...\) Odd terms: \(1^{2}, 3^{2}, 5^{2}...\)

Find the sum of even and odd squared terms

First, let's find the sum of the squares of even and odd terms, respectively. For even terms: \(S_{even} = (-2)^2 + (-4)^2 + \cdots + (-n)^2\) We can rewrite this as: \(S_{even} = 4 + 16 + ... + (n^2)\), we can write n as \(2k\) and now \(S_{even} = 2^2(1^2 + 2^2 + \cdots + k^2)\) For odd terms: \(S_{odd} = 1^2 + 3^2 + 5^2 + \cdots + ((n-1)^2)\) We can rewrite this as: \(S_{odd} = (2 - 1)^2 + (4 - 1)^2 + \cdots = (2\times 1 - 1)^2 + (2\times 2 - 1)^2 + \cdots + (2\times k - 1)^2\).

Apply the arithmetic series formula

To find the sum of these squared terms, we can use the formula for the sum of squares of first 'k' natural numbers: \(\sum_{i=1}^k i^2 = \frac{k(k+1)(2k+1)}{6}\) Now, for even terms: \(S_{even} = 2^2 (\frac{k(k+1)(2k+1)}{6})\) And for odd terms: \(S_{odd} = \frac{k(k+1)(2k+1)}{6}\)

Find the sum of the series and compare the options

Since the series alternates signs, we can find the sum of the series as follows: \(S_{total} = S_{odd} - S_{even} = \frac{k(k+1)(2k+1)}{6} - 2^2 (\frac{k(k+1)(2k+1)}{6})\) Simplifying, we get: \(S_{total} = \frac{k(k+1)(2k+1)}{6} - \frac{4k(k+1)(2k+1)}{6} = -\frac{3k(k+1)(2k+1)}{6} = -\frac{k(k+1)(2k+1)}{2}\) Substituting the value of `k` as \(n = 2k\): \(S_{total} = -\frac{n(n+1)}{2}\) Thus, the correct answer is (c) \(-\frac{n(n+1)}{2}\).

Key concepts

Arithmetic Series

An arithmetic series is a sequence of numbers in which the difference of any two successive members is a constant, known as the common difference. For example, in the series 2, 4, 6, 8, ..., the common difference is 2. The sum of the first 'n' terms of an arithmetic series can be calculated using the formula:

\[S_n = \frac{n}{2} \left(2a_1 + (n-1)d \right)\]where \(S_n\) is the sum of 'n' terms, \(a_1\) is the first term, and \(d\) is the common difference. Arithmetic series play a pivotal role in various mathematical computations and real-world applications, such as calculating loan repayments or predicting sports scores over a season.

In the exercise provided, it's worth noting that the sequence of squared numbers alternates between positive and negative terms, which is a distinctive twist compared to typical arithmetic series. This alternation changes the formula used to calculate the series' sum.

Sum of Squares

The sum of squares is a concept referring to the sum of squared values of terms in a sequence. Specifically, it can refer to the sum of the squares of the first 'n' natural numbers, which is defined by the formula:

\[\sum_{i=1}^n i^2 = \frac{n(n+1)(2n+1)}{6}\]This formula is extremely useful when dealing with polynomial expressions and is derived from mathematical induction. In our problem, understanding the sum of squares allows us to analyze sequences that involve squared terms, such as the sums of squared even and odd numbers. Knowing this general formula can tremendously simplify the process of finding a series' sum by replacing a potentially endless series of calculations with a single expression.

It's important to note that the series in the exercise is not solely based on the sum of squares, because it combines both positive and negative squared terms, effectively modifying the standard summation process.

Even and Odd Numbers

Even and odd numbers are fundamental concepts in mathematics, referring to integers that are divisible by 2 and not divisible by 2, respectively. Every integer is either even or odd, and they have unique properties that affect various mathematical operations. For example:

• Even numbers end with 0, 2, 4, 6, or 8.
• Odd numbers end with 1, 3, 5, 7, or 9.
• The sum of two even numbers or two odd numbers is always even.
• The sum of an even number and an odd number is always odd.

In our exercise, identifying the even and odd terms in the series is crucial to determining the sum. The alternating series of squared even and odd numbers require separate treatment, as the signs of their sums differ due to the series' alternating nature. This characteristic impacts the series' overall sum, highlighting the need to clearly discern between even and odd terms in mathematical sequences.

Mathematical Induction

Mathematical induction is a powerful proof technique used to establish the truth of an infinite number of cases. The process consists of two main steps:

• Base Case: Prove the statement is true for a specific initial case, often for the smallest value (e.g., n=1).
• Inductive Step: Assume the statement is true for a certain case n= 'k' and then prove it is true for n = 'k+1'.

If both steps are successfully completed, by the principle of mathematical induction, the statement is proven to be true for all natural numbers n. This method is particularly woven into deriving formulas for series and sequences, including the sum of squares formula used in our problem. Mathematical induction allows us to generalize the results from finite cases to infinite cases effectively and builds the foundation of much of algebraic theory.