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Sequences & Series
Sequences and Series
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Question

Let Tr{{T_r}} be the rth term of an A.P. whose first term is a and common difference is d. If for some positive integers m, n, mn,Tm=1nandTn=1m,m \ne n,\,\,{T_m} = {1 \over n}\,\,and\,{T_n} = {1 \over m},\, then a - d equals

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Solution

1. Key Concept: The General Term of an Arithmetic Progression (A.P.)

An Arithmetic Progression (A.P.) is a sequence of numbers where the difference between consecutive terms is constant. This constant difference is known as the common difference, denoted by dd. If the first term of an A.P. is aa, then the rr-th term, denoted by TrT_r, can be found using the formula: Tr=a+(r1)dT_r = a + (r-1)d This formula is fundamental for solving problems involving the terms of an A.P.

2. Setting up the Equations from Given Information

We are provided with two crucial pieces of information:

  • The mm-th term of the A.

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