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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, S_p+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, S_m = sum of first m terms.

Then given S_m= S_{m + n} –S_m = S_{m + p} –S_m

S_m = S_{m
+ n} –S_m
&⇒ 2S_m = S_{m
+ 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 S_m = S_{m + 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 t_r =

=

=

&⇒ Sum S_n = = .

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.)

&⇒

&⇒ x^xy^yz^z ≥ . . . (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

a^x, a^x, ….ky times and b^y, b^y, ….kx times .

For all these numbers,

AM =

= .

GM =

= = ….(1)

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

\ (1) becomes or .

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