Problem 22 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

Let \(a_{1}, a_{2}, \ldots ., a_{30}\) be an A.P., \(S=\sum_{i=1}^{30} a_{i}\) and \(\mathrm{T}=\sum_{i=1}^{15} \mathrm{a}_{(2 i-1)^{*}}\) If \(\mathrm{a}_{5}=27\) and \(\mathrm{S}-2 \mathrm{~T}=75\), then \(\mathrm{a}_{10}\) is equal to: [Jan. 09, 2019 (I)] (a) 52 (b) 57 (c) 47 (d) 42

Step-by-Step Solution

Answer

\(a_{10} = 52\), so the answer is (a).

Step 1: Understanding Arithmetic Progression

In an arithmetic progression (A.P.), the general term can be expressed as \( a_n = a_1 + (n - 1)d \), where \( a_1 \) is the first term and \( d \) is the common difference. We need to use this formula to find the required terms.

Step 2: Calculating the 5th Term

We know \( a_5 = 27 \). From the A.P. formula, the 5th term \( a_5 = a_1 + 4d = 27 \). This gives us our first equation: \( a_1 + 4d = 27 \).

Step 3: Determining the Sum S

The sum \( S = \sum_{i=1}^{30} a_i \) of the first 30 terms of the A.P. is given by \[ S = \frac{n}{2}(2a_1 + (n-1)d) \] where \( n = 30 \). So, \[ S = 15(2a_1 + 29d) \].

Step 4: Calculate the Sum T

For the sum \( T = \sum_{i=1}^{15} a_{(2i-1)^*} \), notice \( a_{(2i-1)^*} \) includes terms like \( a_1, a_3, a_5, \ldots, a_{29} \). This is a sequence of odd-positioned terms in the A.P., which forms another A.P. with first term \( a_1 \) and common difference \( 2d \).

Step 5: Evaluating the Series for T

The series \( T \) sums the terms \( a_1, a_3, a_5, \ldots, a_{29} \). The number of terms is 15. So, the sum can be calculated as \[ T = 15 \left( a_1 + (15 - 1) \cdot d \right) / 2 \].

Step 6: Setting the Equation with Given Condition

Given \( S - 2T = 75 \), substitute expressions for \( S \) and \( T \): \[ 15(2a_1 + 29d) - 2 \times \frac{15}{2} (2a_1 + 14d) = 75 \].

Step 7: Simplify the Equation

Simplify the equation from Step 6: \[ 15(2a_1 + 29d) - 15(2a_1 + 14d) = 75 \] This becomes \( 15d = 75 \).

Step 8: Solve for Common Difference d

Solving \( 15d = 75 \) gives \( d = 5 \).

Step 9: Calculate the First Term a1

Substitute \( d = 5 \) back into the equation from Step 2: \[ a_1 + 4 \times 5 = 27 \] Solve to find \( a_1 = 7 \).

Step 10: Find the 10th Term a10

To find \( a_{10} \), use the formula for the nth term of an A.P.: \[ a_{10} = a_1 + 9d = 7 + 9 \times 5 = 52 \].

Key Concepts

Sum of Arithmetic Series

When you have an arithmetic progression, understanding how to find the sum of its series is key. An arithmetic series is the sum of the elements of an arithmetic progression (A.P.). To calculate this sum, you can use the formula: \[ S = \frac{n}{2} (2a_1 + (n-1)d) \] where:

• \( n \) is the number of terms,
• \( a_1 \) is the first term,
• \( d \) is the common difference.

In our exercise, the series comprises 30 terms, so \( n = 30 \). Applying the formula helps you find the total sum of the series. Breaking such problems into parts, like first defining what you know about the series - such as given terms or sums - will lead you to solve for the unknown efficiently.

Common Difference in A.P.

The common difference, represented by \( d \), is critical in arithmetic progressions because it's what makes each sequence term predictable from the last. In any arithmetic sequence, each term is found by adding the common difference to the previous term, which can be denoted as \( a_n = a_{n-1} + d \). For higher terms, the formula converts to \( a_n = a_1 + (n-1)d \). Knowing two terms of the sequence and their position, you can easily solve for this difference.In the provided solution, you are given \( a_5 = 27 \) and later find out \( S - 2T = 75 \), where T is a series sum. Solving this ultimately gives the common difference \( d = 5 \), by setting up and simplifying the equation reflecting the condition given. Understanding how to isolate \( d \) through manipulative algebra is essential in getting to the root of arithmetic sequences.

General Term of A.P.

The general term of an arithmetic progression lets you find any term in the sequence without needing to calculate all preceding terms. This term, \( a_n \), is represented by the formula: \[ a_n = a_1 + (n - 1)d \] This formula is incredibly powerful for large sequences. It tells you by how much the sequence grows term by term using \( d \), and lets you directly compute any specific term. In the exercise you worked with, you used this formula to derive \( a_1 \) from the given \( a_5 = 27 \). After determining \( d \), calculating the 10th term was straightforward as: \[ a_{10} = a_1 + 9d \] This opportune use displays the general term's role in solving parts and wholes of sequences. Mastering the concept enables easier navigation through arithmetic sequences and their applications.