If 100 times the 100^th term of an Arithmetic Progression...

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 |2012|
(A) $-150$
(B) 150 times its $50^{\text {th }}$ term
(C) 150
(D) zero

Key Concepts

Deriving and Solving Equations from Term Relations

Problems involving arithmetic progressions often require setting up equations based on relationships between different terms. By equating expressions derived from the nth term formula for various positions in the sequence, one can solve for unknown parameters such as the first term or the common difference. This technique is central to problem-solving in sequences and series.

Interpreting Special Term Values in AP

Finding a term of the arithmetic progression that satisfies a special condition, like being zero, can provide significant insights into the structure of the sequence. It often indicates a point of symmetry or a balance in the progression, and understanding how such a result arises from the parameters of the AP is important in both theoretical and applied contexts.

Arithmetic Progression (AP)

An arithmetic progression is a sequence of numbers where the difference between consecutive terms is constant. This common difference is a key parameter that defines the sequence, and knowing it allows one to calculate any term in the sequence. APs are fundamental in algebra and are used to model linear relationships in various contexts.

Nth Term Formula

The nth term of an arithmetic progression is given by the formula a? = a + (n - 1)d, where a is the first term and d is the common difference. This formula is crucial because it establishes a direct connection between the index of a term and its value, allowing for the prediction and analysis of the sequence's behavior.

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