First slide

Arithmetic progression

Question

$Let T_{r} be the r^{th} term of an A.P. Whose first term is a and common$ $difference is d . If for some positive integers m, n, m \neq n, T_{m} = \frac{1}{n} and T_{n} = \frac{1}{m}$ $and a - d equals:$

Moderate

A

0
B

1
C

$\frac{1}{m n}$
D

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

Solution

$\begin{array}{l} T_{m} = a + (m - 1) d = \frac{1}{n} \\ T_{n} = a + (n - 1) d = \frac{1}{m} \\ (m - n) d = \frac{1}{n} - \frac{1}{m} = \frac{(m - n)}{m n} \\ So, d = \frac{1}{m n} \\ \begin{aligned} a & = \frac{1}{n} - (m - 1) d = \frac{1}{n} - (m - 1) \frac{1}{m n} = \frac{1}{m n} \\ a - d = 0 \end{aligned} \end{array}$

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