Question

The number of terms in an A.P. is even, the sum of odd terms is 63 and that of even terms is 72 and the last term exceeds the first term by \(16.5\). Find the number of terms : (a) 8 (b) 12 (c) 9 (d) 10

Short answer

Answer: There are 8 terms in the given arithmetic progression.

Step-by-step solution

Identify the formula for the sum of first n terms in A.P.

The formula for the sum of the first n terms in an arithmetic progression having the first term as "a" and the common difference as "d" is given by the formula: \(S_n = \frac{n}{2}(2a + (n-1)d)\)

Find the sums of odd and even terms separately using the formula

Let's first consider the case where we have "n" terms in total, with "\(\frac{n}{2}\)" odd terms and "\(\frac{n}{2}\)" even terms. So, we can write two equations for the sum of odd terms and the sum of even terms using the given information: \(S_\text{odd} = \frac{n}{2}(2a + (n-2)d) = 63\) \(S_\text{even} = \frac{n}{2}(2(a+d) + (n-2)d) = 72\) Subtract the first equation from the second to eliminate "n" and find the relationship between "a" and "d": \(2d = 9\) Therefore, \(d = \frac{9}{2}\).

Use the information that the last term exceeds the first term by 16.5

Since the last term exceeds the first term by 16.5, we can write this relation as: \((a+n-1)d = a + 16.5\)

Test the given options and find the number of terms

We must test each option to see which one satisfies the given conditions. (a) If \(n = 8\), then from the equations \(S_\text{odd} = \frac{8}{2}(2a + 7d) = 63\) and \(S_\text{even} = \frac{8}{2}(2(a + d) + 7d) = 72\), we get \(a = \frac{5}{2}\). Therefore, the last term exceeds the first term by \((\frac{5}{2} + 7 \cdot \frac{9}{2}) - \frac{5}{2} = 16.5\). So, option (a) is correct. (b) If \(n = 12\), then from the equations \(S_\text{odd} = \frac{12}{2}(2a + 11d) = 63\) and \(S_\text{even} = \frac{12}{2}(2(a + d) + 11d) = 72\), we get \(a = -5\) and the last term does not exceed the first term by 16.5. So, option (b) is incorrect. (c) Since the number of terms is even, option (c) is not valid. (d) Since the number of terms is even, option (d) is not valid. So, the correct answer is option (a) with 8 terms in the arithmetic progression.

Key concepts

Sum of an Arithmetic Series

The sum of an arithmetic series is crucial for understanding how to add a sequence of numbers with a common difference. The formula for the sum of the first n terms of an arithmetic progression (A.P.) is:
\[S_n = \frac{n}{2}(2a + (n-1)d)\]
where:

• \(S_n\) is the sum of the first n terms,
• \(n\) is the number of terms,
• \(a\) is the first term, and
• \(d\) is the common difference between the terms.

This formula is derived by taking the average of the first and the last term and multiplying it by the number of terms, because in an A.P., the average of the first and the last term is equal to the average of all the terms. Understanding how to apply this formula can significantly aid in solving problems related to the sum of certain terms in a sequence, particularly when the number of terms is even, and you need to find, for example, the sum of odd or even positioned terms.

Arithmetic Sequence Formula

The arithmetic sequence formula allows us to find any term in the sequence when given the first term and the common difference. The nth term (\(T_n\)) of an arithmetic sequence can be found using the formula:
\[T_n = a + (n-1)d\]
With:

• \(T_n\) representing the nth term,
• \(a\) as the first term,
• \(n\) as the term number, and
• \(d\) as the common difference.

This formula is essential for predicting subsequent numbers in the sequence and is readily used for finding a specific term which is particularly useful when dealing with large sequences, where manually counting each term would be impractical.

Even and Odd Terms in Arithmetic Progression

In an arithmetic progression, even and odd terms can reveal a pattern that simplifies calculations, especially in sequences with an even number of terms. If the number of terms is even, the progression can be split into two equal parts with separate sums of even and odd terms.
For instance, if the number of terms (\(n\)) is even, you can determine the sum of the odd and even terms by considering them as two separate arithmetic sequences. The sum of the odd terms will follow the formula: \[S_\text{odd} = \frac{n}{2}(2a + (n-2)d)\]
and the sum of the even terms is given by: \[S_\text{even} = \frac{n}{2}(2(a+d) + (n-2)d)\]
This method is especially practical when given the sum of odd and even terms, like in the exercise, and needing to find the number of terms or other characteristics of the sequence. Dividing the sequence into odd and even parts can often provide a clearer pathway to the solution.

Quantitative Aptitude

Quantitative aptitude involves the ability to handle numerical and mathematical problems, often within a time constraint, as commonly seen in educational settings and competitive exams. This skill is essential in solving arithmetic progression problems and many other mathematical scenarios.
It requires a good understanding of basic concepts, formulas, and the ability to apply logical thinking and mathematical techniques to solve problems accurately and efficiently. Students with strong quantitative aptitude can often quickly deduce the number of terms in an arithmetic sequence, sum specific segments within the series, and manipulate algebraic expressions to find unknown values, showcasing their problem-solving prowess. Developing this aptitude involves practice with a variety of mathematical problems, learning shortcuts and tricks, and refining the ability to think critically and strategically in quantitative contexts.