Let Tr be the r th term of an A.P. whose first term is a and common difference is d . If for som

step 1 TM is the m's term of the arithmetic series. So it equals the first term plus plus m minus 1 tim

Let Tr be the r th term of an A.P. whose first term is a and...

Let $T_{r}$ be the $r$ th term of an A.P. whose first term is $\underline{a}$ and common difference is $d .$ If for some positive integers $m, n, m \neq n, T_{m}=\frac{1}{n}$ and $T_{n}=\frac{1}{m}$, then $a-d$,
equals
(A) 0
(B)
(C) $\frac{1}{m n}$
(D) $\frac{1}{m}+\frac{1}{n}$

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Key Concepts

Arithmetic Progression

An arithmetic progression (AP) is a sequence of numbers in which each term after the first is obtained by adding a constant value, known as the common difference, to the previous term. This concept is essential because it provides the structure and formula (a + (r-1)*d) that allows us to relate any term in the sequence to its position.

General Term Formula

The general term formula of an AP, a + (r-1)*d, is a key concept as it allows us to explicitly represent any term in the sequence in terms of the first term (a) and the common difference (d). This expression is vital when setting up equations based on given term values.

Systems of Linear Equations

When two terms of the AP are given with specific values, the substitution of these into the general term formula leads to a system of linear equations with the parameters a and d as unknowns. Solving this system is an essential algebraic technique to find relationships between these parameters.

Symmetry in Equations

The presence of symmetry in the given conditions, such as T_m = 1/n and T_n = 1/m, hints at an inherent balance in the relationships between the indices and the term values. Recognizing and exploiting this symmetry can simplify the equation-solving process and lead to insightful conclusions about the parameters of the sequence.

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