SATHEE: Sequences and Series 1 Question 4

Sequences and Series 1 Question 4

4. If $T_{r}$ is the $r$ th term of an $A P$ , for $r = 1, 2, 3, \dots$ . . If for some positive integers $m$ and $n$ , we have $T_{m} = \frac{1}{n}$ and $T_{n} = \frac{1}{m}$ , then $T_{m n}$ equals

$(1998, 2 M)$

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Answer:

Correct Answer: 4. (6)

Solution:

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

and

On subtracting Eq. (ii) from Eq. (i), we get

$\begin{aligned} (m - n) d = \frac{1}{n} - \frac{1}{m} & = \frac{m - n}{m n} \\ \Rightarrow d & = \frac{1}{m n} \end{aligned}$

Again, $T_{m n} = a + (m n - 1) d = a + (m n - n + n - 1) d$

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

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Sequences and Series 1 Question 4

4. If Tr is the r th term of an AP, for r=1,2,3,…. . If for some positive integers m and n, we have Tm=1n and Tn=1m, then Tmn equals

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4. If $T_{r}$ is the $r$ th term of an $A P$ , for $r = 1, 2, 3, \dots$ . . If for some positive integers $m$ and $n$ , we have $T_{m} = \frac{1}{n}$ and $T_{n} = \frac{1}{m}$ , then $T_{m n}$ equals