If a is the first term, d the common difference and Sk the...

If $a$ is the first term, $d$ the common difference and $S_{k}$ the sum to $k$ terms of an A.P., then for $\frac{S_{k x}}{S_{x}}$ to be inde- pendent of $x$
(A) $a=2 d$
(B) $a=d$
(C) $2 a=d$
(D) None of these

Key Concepts

Comparing Coefficients

The method of comparing coefficients is used to determine the values of unknown parameters in expressions by equating the coefficients of like terms. This technique is crucial when ensuring that an expression holds true for all values of a variable, effectively eliminating the variable's presence from the final expression.

Independence of a Variable in a Rational Expression

When an expression, such as a ratio involving sums, is required to be independent of a variable, it means that the expression must simplify to a constant that does not change with the variable. Achieving this typically involves finding conditions that eliminate the variable from the outcome, ensuring that any dependence on the variable cancels out.

Arithmetic Progression

An arithmetic progression (AP) is a sequence of numbers in which each term after the first is obtained by adding a fixed constant, known as the common difference. This concept is fundamental because it establishes a linear pattern within sequences, allowing for predictable behavior and analysis of series.

Sum Formula for Arithmetic Progression

The sum of the first n terms of an arithmetic progression is given by the formula S? = (n/2)[2a + (n-1)d], where a is the first term and d is the common difference. This formula is essential for evaluating the total of a number of terms and is extensively used in various problems involving series summations.

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