Q18 of 47 Page 1

If the mth term of an A. P. is 1/n and nth term is 1/m then show that its (mn)th term is 1.

Given that am = 1/n and an = 1/m

We have to show that amn = 1.


We know that in an AP with first term a and common difference d, the nth term (or the general term) is given by an = a + (n – 1) d.


∴ am = a + (m – 1) d and an = a + (n – 1) d


⇒ 1/n = a + (m – 1) d … (1)


⇒ 1/m = a + (n – 1) d … (2)


Solving (1) and (2),


⇒ (a – a) + (md – d – nd + d) = 1/n – 1/m


⇒ (m-n) d = 1/n – 1/m


⇒ (m – n) d = (m – n) / mn


∴ d = 1/mn


Substituting d value in (1),


⇒ a + (m – 1) (1/mn) = 1/n


⇒ a + 1/n – 1/mn = 1/n


⇒ a = 1/mn + 1/n – 1/n


∴ a = 1/mn


Now, amn = a + (mn – 1) d


Substituting a and d values,


⇒ amn = (1/mn) + (mn – 1) (1/mn)


⇒ amn = (1/mn) + 1 – (1/mn)


∴ amn = 1


Ans. Hence from the given conditions we showed that amn = 1.


More from this chapter

All 47 →
31

In a hospital used water is collected in a cylindrical tank of diameter 2 m and height 5 m. After recycling, this water is used to irrigate a park of hospital whose length is 25 m and breadth is 20 m. If tank is filled completely then what will be the height of standing water used for irrigating the park. Write your views on recycling of water.

 10

Draw a line segment of length 7 cm and divide it internally in the ratio 2: 3.

 19

A metallic solid sphere of radius 10.5 cm is melted and recasted into smaller solid cones, each of radius 3.5 cm and height 3 cm. How many cones will be made?

 20

From the top of a 7 m high building, the angle of elevation of the top of a tower is 60° and the angle of depression of its foot is 45°. Find the height of the tower.