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Arithmetic Progression

The following...

Question

The following is known about an AP:

$T_{m} = \frac{1}{n}, T_{n} = \frac{1}{m}$

Find the value of $S_{m n}$ . [4 MARKS]

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Solution

Formula: 1 Mark
Application: 2 Marks
Answer: 1 Mark

Let a and d be the first term and the common difference of the AP. We have:

$T_{m} = a + (m - 1) d = \frac{1}{n}$

$T_{n} = a + (n - 1) d = \frac{1}{m}$

$\Rightarrow T_{m} - T_{n} = (m - n) d = \frac{1}{n} - \frac{1}{m}$

$\Rightarrow (m - n) d = \frac{m - n}{m n}$

$\Rightarrow d = \frac{1}{m n}$

Using this value of d in the any of the two initial relations, we obtain $a = \frac{1}{m n}$ .

$S_{m n} = (\frac{m n}{2}) (2 a + (m n - 1) d)$

$= (\frac{m n}{2}) {\frac{2}{m n} + (m n - 1) (\frac{1}{m n})}$

$= \frac{m n + 1}{2}$

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