Question

The sum of first 3 terms of hp is 33/40 and 1st term is 1/2.Find progression

Answer 1

The Harmonic Progression can either be 1/2, 13/8, -13/10... or it can be 1/2, 1/5, 1/8...

Given:

The sum of the first 3 terms of the HP is 33/40.

The first term of the HP is 1/2.

To Find:

The Harmonic Progression.

Solution:

The first term of the HP is 1/2.

$\boldsymbol{Let\:the\:1st\:3\:terms\:of\:the\:HP\:be\\:\frac{1}{2} ,\:\frac{1}{2+d}\: ,\frac{1}{2+2d} }$

→ The sum of the first 3 terms of the HP is 33/40.

                   $\boldsymbol{\therefore \frac{1}{2}+ \frac{1}{2+d} +\frac{1}{2+2d} =\frac{33}{40} }\\\\\boldsymbol{\therefore \frac{(2+d)(2+2d)+2(2+2d)+2(2+d)}{2(2+d)(2+2d)} =\frac{33}{40} }\\\\\boldsymbol{\therefore \frac{(2d^{2}+6d+4 )+(4d+4 )+(2d+4 )} {2(2d^{2}+6d+4 )}=\frac{33}{40} }\\\\\boldsymbol{\therefore \frac{(2d^{2}+12d+12 )} {(2d^{2}+6d+4 )}=\frac{33}{20} }\\\\\boldsymbol{\therefore \frac{(d^{2}+6d+6)} {(d^{2}+3d+2 )}=\frac{33}{20} }$

                $\boldsymbol{\therefore 33d^{2} +99d+66=20d^{2} +120d+120}\\\\\boldsymbol{\therefore 13d^{2} -21d-54=0}\\\\\boldsymbol{\therefore (13d+18)(d-3)=0}\\\\\boldsymbol{\therefore d=-18/13\:or\:3}$

→ Now, finding the Harmonic Progressions:

   (1.) For d = -18/13:

      (i) (2 + d) = 2 - (18/13) = 8/13

      (ii) (2 + 2d) = 2 - 2(18/13) = -10/13

    $\boldsymbol{Therefore\:the\:HP\:would\:be\\:\frac{1}{2} ,\:\frac{1}{2+d}\: ,\frac{1}{2+2d} ....}\\\\\boldsymbol{Therefore\:the\:HP\:would\:be\\:\frac{1}{2} ,\:\frac{1}{(8/13)}\: ,\frac{1}{(-10/13)} ....}\\\\\boldsymbol{Therefore\:the\:HP\:would\:be\\:\frac{1}{2} ,\:\frac{13}{8}\: ,\frac{-13}{10}.... }$

   (2.) For d = 3:

        (i) (2 + d) = 2 + 3 = 5

        (ii) (2 + 2d) = 2 + 2(3) = 8

    $\boldsymbol{Therefore\:the\:HP\:would\:be\\:\frac{1}{2} ,\:\frac{1}{2+d}\: ,\frac{1}{2+2d} ....}\\\\\boldsymbol{Therefore\:the\:HP\:would\:be\\:\frac{1}{2} ,\:\frac{1}{5}\: ,\frac{1}{8} ....}$

Therefore the Harmonic Progression can either be 1/2, 13/8, -13/10... or it can be 1/2, 1/5, 1/8...

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Answer 2

Answer:- The Harmonic Progression can be 1/2, 13/8, -13/10... or it can be 1/2, 1/5, 1/8...

Given:

The sum of the first 3 terms of the HP is 33/40.

The first term of the HP is 1/2.

To Find the Harmonic Progression we need to do is:-

Solution:

The first term of the HP is 1/2.

The sum of the first 3 terms of the HP is 33/40.

Now, I am finding the Harmonic Progressions: -

For d = -18/13:

(i) (2 + d) = 2 - (18/13) = 8/13

(ii) (2 + 2d) = 2 - 2(18/13) = -10/13

For d = 3

(i) (2 + d) = 2+3 = 5

(ii) (2 + 2d) = 2+2(3) = 8

Therefore, the Harmonic Progression can be 1/2, 13/8, -13/10... or it can be 1/2, 1/5, 1/8...

A Harmonic Progression (HP) is defined as a sequence of real numbers which is determined by taking the reciprocals of the arithmetic progression i.e., AP that does not contain 0.

In harmonic progression, any term in the sequence is considered as the harmonic means of its two neighbors.

The sum of HP is also the reciprocal of the sum of the AP.

To know more about the given topic please go through the following

Link1:- https://brainly.in/question/2054504?

Link2:- https://brainly.in/question/4925969?

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