Problem 145
Question
The sum of the first \(\mathrm{n}\) terms of the series \(\mathrm{I}^{2}+2 \cdot 2^{2}\) \(+3^{2}+2 \cdot 4^{2}+5^{2}+2 \cdot 6^{2}+\ldots\) is \(\frac{n(n+1)^{2}}{2}\) when \(n\) is even. When \(n\) is odd the sum is \(\quad\) [2004] (A) \(\frac{3 n(n+1)}{2}\) (B) \(\frac{n^{2}(n+1)}{2}\) (C) \(\frac{n(n+1)^{2}}{4}\) (D) \(\left[\frac{n(n+1)}{2}\right]^{2}\)
Step-by-Step Solution
Verified Answer
The answer is B: \(\frac{n^2(n+1)}{2}\).
1Step 1: Understanding the Series
The series given is a combination of terms of the form \( n^2 \) and \( 2m^2 \) depending on whether the term index is odd or even. Specifically, the series alternates with terms \( n^2 \) for odd \( n \) and \( 2n^2 \) for even \( n \).
2Step 2: Finding the Expression for Even Terms
For even \( n \), the formula for the sum of the series is provided as \( \frac{n(n+1)^2}{2} \). This means that every pair of odd and even terms contribute to this sum, completing the series for even numbers.
3Step 3: Summing the Series for Odd Terms
When \( n \) is odd, the last term in the series will be of the form \( n^2 \), which is not doubled as it would be when \( n \) is even. Thus, the sum needs to accommodate this condition.
4Step 4: Deriving the Sum for Odd Terms
Out of \( n \) terms, the sum for the even number of terms (i.e., \( n-1 \)) can be given as \( \frac{(n-1)n^2}{2} \) according to the formula for even \( n \). We need to add the last term \( n^2 \) to this result, as it is odd and not doubled.
5Step 5: Calculating the Odd Sum
Thus, the total sum when \( n \) is odd is \[ \frac{(n-1)(n+1)^2}{2} + n^2. \] Simplifying, we get \[ \frac{n(n+1)(2n+1)}{6} + n^2. \] Reviewing our set of possible answers, the correct one needs to be found.
6Step 6: Matching the Correct Answer
After simplifying and comparing with the provided options, the correct answer matching the condition where \( n \) is odd, is option B: \( \frac{n^2(n+1)}{2} \).
Key Concepts
Alternating SeriesSum of SeriesEven and Odd Terms
Alternating Series
In mathematics, an alternating series is a sequence of numbers where the terms switch signs. For example, just like in the given series problem, some terms appear as simple squares, while others may be doubled, depending on their position. Alternating series are significant because they can converge, meaning that even if you keep adding more and more terms, the total still approaches a specific value. Alternating means that although the terms might build on top of each other, some will add while others might subtract, all based on their positioning. This unique property is particularly useful in certain mathematical computations and real-world applications, where balancing positive and negative increments is essential.
Sum of Series
Summing a series involves adding together all the consecutive terms to compute a total. In the exercise provided, there are formulas for both even and odd terms. When you sum up the series where n is even, you use a specific formula \( \frac{n(n+1)^2}{2} \). This formula takes into account alternating terms to give a neat and simple solution.
An important characteristic of solving series is identifying whether you need to approach with a single formula or if separate ones are necessary for even and odd terms. Understanding this distinction helps in simplifying complex series into manageable parts, like breaking down the series into smaller, digestible results. Furthermore, distinguishing the contribution of specific terms, such as last terms in odd series, becomes vital in ensuring your sum accurately represents all components.
An important characteristic of solving series is identifying whether you need to approach with a single formula or if separate ones are necessary for even and odd terms. Understanding this distinction helps in simplifying complex series into manageable parts, like breaking down the series into smaller, digestible results. Furthermore, distinguishing the contribution of specific terms, such as last terms in odd series, becomes vital in ensuring your sum accurately represents all components.
Even and Odd Terms
The differentiation between even and odd terms is crucial in the context of series problems. An even term is one that falls in a position described by multiples of two (2, 4, 6, etc.), while an odd term is not (1, 3, 5, etc.).
In this series, odd terms consist of simple squared numbers, and the even terms are doubled squares. Such categorization affects how their sums are computed. For an odd number of terms, the series' last term is treated differently compared to its neighboring terms.
In this series, odd terms consist of simple squared numbers, and the even terms are doubled squares. Such categorization affects how their sums are computed. For an odd number of terms, the series' last term is treated differently compared to its neighboring terms.
- Even terms: These are larger because they are doubled (e.g., 2 \( \cdot 4^2 \)).
- Odd terms: Simply as squared numbers (e.g., \( 3^2 \)).
Other exercises in this chapter
Problem 139
Fifth term of a GP is 2, then the product of its 9 terms is : \(\quad[2002]\) (A) 256 (B) 512 (C) 1024 (D) None of these
View solution Problem 142
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View solution Problem 147
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In a geometric progression consisting of positive terms, each term equals the sum of the next two terms. Then the common ratio of this progression equals (A) \(
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