The sixth term of an A.P. is equal to 2 . The value of the...

The sixth term of an A.P. is equal to 2 . The value of the common difference of the A.P. which makes the product $a_{1} a_{4} a_{5}$ greatest, is
(A) $\frac{8}{5}$
(B) $\frac{2}{3}$
(C) $\frac{3}{5}$
(D) $\frac{3}{4}$

Key Concepts

Arithmetic Progression (A.P.)

An Arithmetic Progression is a sequence of numbers in which the difference between any two successive terms is constant. This key concept underpins a wide range of problems in algebra and number theory and provides a systematic way to analyze patterns and relationships between terms in a sequence.

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 as it allows one to relate any term in the sequence to the first term and the common difference, and it is particularly useful when constraints about specific term values are provided.

Product Optimization in Sequences

Product optimization involves finding the value of a variable (in this case, the common difference) that maximizes or minimizes a product of certain terms of a sequence. In problems where terms of an A.P. are related by a constraint, expressing the product in terms of one variable and finding its maximum is a common application of optimization techniques.

Differentiation for Optimization

Differentiation is the method used in calculus to find the maximum or minimum values of functions. By expressing the product to be optimized as a function of a single variable, taking its derivative, and setting the derivative equal to zero, one can solve for the value that gives the optimal product. This technique is fundamental in solving a wide array of optimization problems in mathematics.

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