Problem 156 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If 100 times the \(100^{\text {th }}\) term of an Arithmetic Progression 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 of the AP is zero.

Step 1: Identify the General Term of an AP

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

Step 2: Express the Given Condition

The exercise states that 100 times the 100th term equals 50 times the 50th term: \( 100 imes a_{100} = 50 imes a_{50} \). Substitute the general term equation to get: \[ 100(a + 99d) = 50(a + 49d) \].

Step 3: Simplify the Equation

Divide both sides of the equation by 50 to simplify: \[ 2(a + 99d) = a + 49d \].

Step 4: Solve for a in terms of d

Distribute and solve for \( a \): \( 2a + 198d = a + 49d \). Rearrange to get \( a = -149d \).

Step 5: Find 150th Term of the AP

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

Step 6: Simplify the Expression

Simplify the expression to see that \( a_{150} = 0 \).

Key Concepts

General Term of AP

In an Arithmetic Progression (AP), every term after the first is formed by adding a constant difference to the preceding term. This leads us to the formula for any term in an AP series, which is known as the 'general term'. The general term is expressed as \( a_n = a + (n-1)d \). Here, \( a \) is the first term of the sequence, \( n \) corresponds to the term's position in the sequence, and \( d \) represents the common difference between consecutive terms.
This formula allows you to calculate any term in the sequence provided you know the initial term and the common difference. Understanding this concept is crucial as it forms the foundation for solving various problems related to sequences.

Common Difference in AP

The common difference \( d \) is a pivotal part of an Arithmetic Progression, signifying the consistent interval between consecutive terms. To comprehend how it influences an AP, imagine the sequence of numbers as steps equally spaced apart.
This constant difference is the key to determining the progression's behavior; whether it continuously ascends, descends, or remains static if \( d \) equals zero.

• If \( d \) is positive, the sequence ascends or increases.
• If \( d \) is negative, the sequence descends or decreases.
• If \( d \) is zero, the sequence remains constant at every term.

Grasping the concept of the common difference is essential, especially when manipulating or establishing relationships between terms in problems like the original exercise provided.

Sequence Solving through Substitution

Solving arithmetic sequence problems often involves substituting known relationships into the general term formula. In our specific exercise, we identified relationships between multiple terms using given conditions about the sequence.
This step-by-step approach involves:

• Substituting terms within the formula (e.g., \( a_{100} \) and \( a_{50} \)) with expressions derived from the general term.
• Simplifying these expressions by performing algebraic operations, such as distributing and rearranging terms.
• Solving for unknowns (like the first term \( a \) or the common difference \( d \)) whenever possible.

This method is not only beneficial for solving specific problems but also for understanding deeper relationships within an AP. Sequence solving essentially transforms the problem into more manageable algebra before deriving an accurate answer.