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Let $T_r$ be the rth term of an A.P. whose first term is a and common difference is $d$. If for some positive integers $m, n, m \neq n, T_m=\frac{1}{n}$ and $T_n=\frac{1}{m}$, then $a-d$ equals
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$$
T_m=\frac{1}{n}=a+(m-1) d
$$
and $T_n=\frac{1}{m}=a+(n-1) d$
from (1) and (2) we get $a=\frac{1}{m n}, \quad d=\frac{1}{m n}$ Hence $\mathrm{a}-\mathrm{d}=0$
T_m=\frac{1}{n}=a+(m-1) d
$$
and $T_n=\frac{1}{m}=a+(n-1) d$
from (1) and (2) we get $a=\frac{1}{m n}, \quad d=\frac{1}{m n}$ Hence $\mathrm{a}-\mathrm{d}=0$
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