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Variable Separable Method

Let Tr be t...

Question

Let $T_{r}$ be the $r^{t h}$ term of an A.P., for $r = 1, 2, . . .$ If for some positive integers $m$ and $n, T_{m} = \frac{1}{n}; T_{n} = \frac{1}{m}$ then find $T_{m n}$ .

\frac{1}{m n}

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\frac{1}{m} + \frac{1}{n}

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Solution

The correct option is D $1$
$T_{m} = \frac{1}{n}$ (given)
$a + (m - 1) d = \frac{1}{n} ⟶ 1$
$T_{n} = \frac{1}{m}$ (given)
$a + (n - 1) d = \frac{1}{m} ⟶ 2$
Substract equation 2 from 1, we get
$(m - 1) d - (n - 1) d = \frac{1}{n} - \frac{1}{m}$
$d [m - 1 - n + 1] = \frac{m - n}{m n}$
$d [m - n] = \frac{m - n}{m n}$
$d = \frac{1}{m n} ⟶ 3$
Put value of d in equation 1
$a + (m - 1) \frac{1}{m n} = \frac{1}{n}$
$\frac{a m n + (m - 1)}{m n} = \frac{1}{n}$
$a m n + m - 1 = m$
$a = \frac{1}{m n} ⟶ 4$
$T_{m n} = a + (m n - 1) d$
$T_{m n} = \frac{1}{m n} + (m n - 1) \frac{1}{m n}$ (From equation 3 & 4)
$T_{m n} = \frac{1 + m n - 1}{m n} = \frac{m n}{m n}$
$T_{m n} = 1$

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Let Tr be the rth term of an A.P., for r=1,2,... If for some positive integers m and n,Tm=1n;Tn=1m then find Tmn.

Let $T_{r}$ be the $r^{t h}$ term of an A.P., for $r = 1, 2, . . .$ If for some positive integers $m$ and $n, T_{m} = \frac{1}{n}; T_{n} = \frac{1}{m}$ then find $T_{m n}$ .