If 100 times the 100^th term of an A P with non-zero common...

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

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Key Concepts

Arithmetic Progression

An arithmetic progression (AP) is a sequence in which each term is obtained by adding a constant, known as the common difference, to the preceding term. This concept is fundamental in the study of sequences and series because it provides a predictable pattern that can be expressed using a simple relationship.

nth Term Formula

The nth term of an arithmetic progression is calculated using the formula a? = a? + (n - 1)d, where a? is the first term and d is the common difference. This formula is essential for identifying any term in the sequence and forms the basis for setting up equations that relate different terms.

Setting Up and Solving Equations in an AP

In problems involving arithmetic progressions, conditions that relate different terms can be translated into equations using the nth term formula. Solving these equations allows you to determine unknown parameters of the sequence, such as the first term or the common difference, and to find other specific term values.

Common Difference

The common difference is the constant amount that is added to each term in an arithmetic progression to generate the next term. It is a critical component of the sequence's structure and is key to understanding how the sequence evolves and how individual terms relate to one another.

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