Question

(b) If mth and nth term of an A.P. are 1/n and 1/m respectively, then find its
mnth
term.
​

Answer 1

EXPLANATION.

⇒ Mth terms of an A.P. = 1/n.

⇒ Nth terms of an A.P. = 1/m.

As we know that,

General terms of an A.P.

⇒ Tₙ = a + (n - 1)d.

Using this formula in equation, we get.

⇒ T(m) = a + (m - 1)d = 1/n. - - - - - (1).

⇒ Tₙ = a + (n - 1)d = 1/m. - - - - - (2).

Subtract the equation (1) & (2), we get.

⇒ (m - n)d = 1/n - 1/m.

⇒ (m - n)d = m - n/nm.

⇒ d = 1/nm.

Put the value of d = 1/nm in equation (1), we get.

⇒ a + (m - 1)(1/nm) = 1/n.

⇒ a + [m/nm - 1/nm] = 1/n.

⇒ a + [1/n - 1/nm] = 1/n.

⇒ a - 1/nm = 1/n - 1/n.

⇒ a - 1/nm = 0.

⇒ a = 1/nm.

To find :

⇒ T(mn terms).

Put the value of a and d = 1/nm in equation, we get.

⇒ T(mn) = 1/nm + (mn - 1)(1/nm).

⇒ T(mn) = 1/nm + [mn/nm - 1/nm].

⇒ T(mn) = 1/nm + 1 - 1/nm.

⇒ T(mn) = 1.

                                                                                                                           

MORE INFORMATION.

Supposition of an A.P.

(1) = Three terms as : a - d, a, a + d.

(2) = Four terms as : a - 3d, a - d, a + d, a + 3d.

(3) = Five terms as : a - 2d, a - d, a, a + d, a + 2d.

Answer 2

Answer:

Given :-

If mth and nth term of an A.P. are 1/n and 1/m respectively,

To Find :-

mnth term

Solution :-

For mth term

$\bf \: a_{m} = a(m - 1)d$

$\sf \: a _{m} = a + md + d = \dfrac{1}{m} (1)$

For nth term

$\bf \: a_n = a(n - 1)d$

$\sf \: a_{n} = \: a + nd - d = \dfrac{1}{n} (2)$

Now,

According to the question

$\sf \bigg(m - n \bigg)d = \dfrac{1}{m} - \dfrac{1}{n}$

$\sf \bigg(m - n \bigg)d= \dfrac{m-n}{mn}$

$\sf \: \: d= \dfrac{1}{mn}$

Now,

$\sf a + (m-1)\bigg(\dfrac{1}{mn}\bigg) = \dfrac{1}{n}$

$\sf a + \bigg(\dfrac{1}{n} - \dfrac{1}{mn} \bigg) = \dfrac{1}{n}$

$\sf \: a - \dfrac{1}{mn} = \dfrac{1}{n} - \dfrac{1}{n}$

$\sf \: a \: - \dfrac{1}{mn} = \dfrac{1 - 1}{n}$

$\sf \: a - \dfrac{1}{mn} = 0$

T(mn) = 1/mn + (mn/nm - 1/nm)

T(mn) = 1/mn + (1/1 - 1/nm)

T(mn) = 1/mn + 1 - 1/nm

T(mn) = 1/mn + 1 - 1/nm

T(mn) = 1