Problem 43 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If 100 times the \(100^{\text {th }}\) term of an \(A P\) with non-zero common difference equals the 50 times its \(50^{\text {th }}\) term, then the \(150^{\text {th }}\) term of this \(A P\) is (A) \(-150\) (B) 150 times its \(50^{\text {th }}\) term (C) 150 (D) zero

Step-by-Step Solution

Answer

The 150th term is zero (Option D).

Step 1: Understanding the Problem

We need the 100th term and the 50th term of an arithmetic progression (AP) and we know 100 times the 100th term equals 50 times the 50th term. We'll find the 150th term based on this information.

Step 2: Identify the General Term

The general term of an AP is given by the formula \(a_n = a + (n-1)d\), where \(a\) is the first term and \(d\) is the common difference.

Step 3: Formulate the Given Equation

The 100th term, \(a_{100} = a + 99d\), and the 50th term, \(a_{50} = a + 49d\). According to the problem: \(100(a + 99d) = 50(a + 49d)\).

Step 4: Solve the Equation

Calculate the above equation: \[100a + 9900d = 50a + 2450d\]Subtracting 50a and 2450d from both sides:\[50a + 7450d = 0\]This simplifies to \[a = -149d\].

Step 5: Find the 150th Term

Use the general term formula to find the 150th term:\[a_{150} = a + 149d\]Substitute \(a = -149d\):\[a_{150} = -149d + 149d = 0\].

Step 6: Determine the Correct Answer

The 150th term is 0, which corresponds to option (D).

Key Concepts

General Term of an AP

In arithmetic progressions (APs), understanding how to find terms is essential. Each term in an AP can be calculated using the general term formula. This formula is useful for identifying any specific term in the sequence, given the first term and the common difference. The general term of an AP is expressed as \(a_n = a + (n-1)d\). Here:

• \(a\): The first term of the arithmetic progression.
• \(n\): The position of the term in the sequence.
• \(d\): The common difference between consecutive terms.

This means that to find the nth term, we start from the first term and add the common difference \((n-1)\) times. Let's say you have an AP where the first term \(a = 3\) and the common difference \(d = 2\). To find the 5th term, you would substitute into the formula: \[a_5 = 3 + (5-1)\times2 = 3 + 8 = 11\]So, the 5th term would be 11. This systematic approach helps in solving more complex problems, like the one in the exercise, where the specific terms must satisfy a given condition.

Common Difference

The common difference in an arithmetic progression (AP) is what defines the interval between consecutive terms. It is usually represented by the letter \(d\) and plays a fundamental role in the structure of an AP. Every term after the first is derived by adding the common difference to the previous term.To find the common difference in an AP, simply subtract any term from the term that follows it. If an AP starts with \(5, 9, 13, 17\), then the common difference \(d\) is \(9 - 5 = 4\). This same difference repeats throughout the sequence.Understanding \(d\) is crucial when solving AP problems, as it helps in using formulas effectively. In our exercise, identifying that \(a = -149d\) demonstrates the relationship between the first term and the common difference. This relationship ensures that calculations for any term in the sequence are consistent and aligned with the given conditions.

Nth Term Formula

The nth term formula is the backbone of understanding arithmetic progressions. It allows the determination of any term's value within a sequence. The formula is \(a_n = a + (n-1)d\), enabling prediction of the sequence's behavior without listing all terms.Using the nth term formula, you can quickly and accurately compute the value of large-term positions. For example, if the first term \(a = 1\) and the common difference \(d = 3\), you can calculate the 100th term as follows: \[a_{100} = 1 + (100-1)\times3 = 1 + 297 = 298\]In problems requiring relationships between terms, as shown in our exercise, this formula becomes indispensable. It allowed us to express the 100th and 50th terms, leading to the condition \(100(a + 99d) = 50(a + 49d)\), ultimately determining the 150th term. Mastery of this formula enhances your ability to tackle varied AP problems efficiently.