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Solved Subjective Question on Progression and Series Set 3

Posted on - 21-02-2017

JEE Math P And S

IIT JEE

Example 1

The p th term of an A.P. is a and q th term is b. Prove that sum of it's
(p + q) terms is

Solution:

Let x be the first term and y be the c . d of A.P. .

a = x + (p – 1 ) d

b = x + ( q – 1)d


&⇒
d = . . . . . (1)

so, x = a -

=

Hence, Sp+q = .

Example 2

If the sum of m terms of an arithmetical progression is equal to the sum of either the next n terms or the next p terms, prove that
(m + n)= (m +p) .

Solution:

Let first term = a c. d = d, Sm = sum of first m terms.

Then given Sm = Sm + n –Sm = Sm + p –Sm

Sm = Sm + n –Sm
&⇒
2Sm = Sm + n


&⇒
2

2a [2m –m –n] = d [m2 + n2 + 2mn –m –n –2m2 + 2m]

2a [m –n] = d [ n2 –m2 + 2mn + m –n ] ….(1).

also 2 Sm = Sm + p


&⇒
2a [m –p] = d [p2 –m2 + 2mp +m –p] …
.(2).

from (1) and (2)


&⇒


&⇒


&⇒


&⇒


&⇒

Example 3

Sum of the series to n terms

Solution:

Here tr =

=

=


&⇒
Sum Sn = = .

Example 4

For positive real numbers x, y, z prove that

Solution:

Let x, y, z be three numbers with weights x, y, z respectively. Then.

(weighted A. M. ≥ weighted G. M.)

. . . (A)

Again let be three numbers with weights x, y, z respectively then

(weighted A.M ≥ weighted G.M.)


&⇒


&⇒
xxyyzz ≥ . . . (B)

Using (A) and (B) , we get the result .

Example 5

If a, b, x, y are positive natural numbers such that then prove that .

Solution:

Consider the positive numbers

ax, ax, ….ky times and by, by, ….kx times .

For all these numbers,

AM =

= .

GM =

= = ….(1)

As , , i.e. x + y = xy

\ (1) becomes or .

 
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