Problem 9 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

Let the sum of the first three terms of an A. P, be 39 and the sum of its last four terms be 178 . If the first term of this A.P. is 10 , then the median of the A.P. is : [Online April 10, 2015] (a) 28 (b) \(26.5\) (c) \(29.5\) (d) 31

Step-by-Step Solution

Answer

The median of the A.P. is 29.5 (Option c).

Step 1: Define Given Information

We are given an arithmetic progression (A.P.) where the first term \( a = 10 \). The sum of the first three terms is 39, and the sum of the last four terms is 178.

Step 2: Express First Three Terms

Since it's an A.P., the terms can be represented as \( a, a+d, a+2d \). The sum of these terms is 39:\[a + (a+d) + (a+2d) = 3a + 3d = 39\]Since \( a = 10 \), substitute into the equation:\[3(10) + 3d = 39\]\[30 + 3d = 39\]Solve for \( d \):\[3d = 9 \quad \Rightarrow \quad d = 3\]

Step 3: Determine Number of Terms

Let the total number of terms in the A.P. be \( n \). The last four terms can then be expressed as \((a + (n-4)d), (a + (n-3)d), (a + (n-2)d), (a + (n-1)d)\). Their sum is 178:\[4a + (4n-10)d = 178\]Substitute the values \( a = 10 \) and \( d = 3 \):\[4(10) + (4n-10)(3) = 178\]Simplify:\[40 + 12n - 30 = 178\]\[12n + 10 = 178\]\[12n = 168\]\[n = 14\]

Step 4: Find Median of A.P.

Since there is an even number of terms (\( n = 14 \)), the median is the average of the 7th and 8th terms. They can be expressed as:\[7th \ term: a + 6d = 10 + 6(3) = 28\]\[8th \ term: a + 7d = 10 + 7(3) = 31\]The median is the average of these two values:\[\text{Median} = \frac{28 + 31}{2} = 29.5\]

Key Concepts

Sum of Terms

In an arithmetic progression (A.P.), the sum of terms is a crucial concept, especially when trying to calculate the values of the progression. An A.P. is a sequence of numbers in which the difference between consecutive terms, known as the common difference, remains constant. For any arithmetic sequence, the sum of the first few terms can be calculated promptly.

• Sum Formula: The sum of the first n terms of an arithmetic sequence can be determined using the formula \[ S_n = \frac{n}{2} \cdot (2a + (n-1)d) \]where \( S_n \) is the sum of the first n terms, \( a \) is the first term, and \( d \) is the common difference.
• Example: In the given problem, the sum of the first three terms is 39. By knowing this, along with the information of the first term \( a = 10 \), we can easily set up the equation to discover the common difference.

Understanding how to find the sum is often one of the steps in solving more complex arithmetic progression problems, like determining the number of total terms or the median of the sequence.

Median in Sequences

The median is the middle value of a data set or sequence when arranged in order. In arithmetic progressions, finding the median can be slightly more interesting due to the regular spacing of terms.

• Odd Number of Terms: If the sequence has an odd number of terms, the median is simply the middle term. For example, in a sequence with 5 terms, the 3rd term is the median.
• Even Number of Terms: For sequences with an even number of terms, such as the A.P. in our problem with 14 terms, the median is the average of the two middle terms. Here, it was determined by averaging the 7th and 8th terms, resulting in a median of 29.5.

This concept highlights the importance of identifying the position of terms, especially when dealing with ordered sequences like arithmetic progressions.

Number of Terms in Sequence

Knowing the number of terms in an arithmetic progression can help solve a variety of related problems. The total number of terms defines the span of the sequence and can impact computations such as the sum and the median.

• General Formula: If you know the last term of an A.P. and you need to determine the number of terms \( n \), you can use \[ T_n = a + (n-1)d \]where \( T_n \) is the last term, \( a \) is the first term, and \( d \) is the common difference.
• Applying the Concept: In the original problem, by using the information about the sum of the last four terms being 178 and solving equation \[ 4a + (4n-10)d = 178 \], we were able to find the total number of terms, which is 14.

Understanding how to manipulate these key variables is essential for tackling more in-depth problems within arithmetic sequences.