JEE progressions

Algebra Level 2

Given that

$a,b,c$ form an arithmetic progression,
$b,c,d$ form a geometric progression, and
$c,d,e$ form a harmonic progression,

what progression will $a,c,e$ form?

Harmonic progression Arithmetic Progression None of these choices Geometric Progression

2 solutions

Rohit Ner
May 27, 2015

$2b=a+c\\{ c }^{ 2 }=bd\\\Rightarrow 2\frac { { c }^{ 2 } }{ d } =a+c\\d=2\frac { ce }{ c+e } \\\Rightarrow 2\frac { { c }^{ 2 } }{ \frac { 2ce }{ c+e } } =a+c\\{ c }^{ 2 }=ae$

Hence $a,c,e$ will form a geometric progression.

Nice method, Rohit. :)

Brian Charlesworth - 6 years ago

@Brian Charlesworth Sir, actually this is one question that all class 11 students in India solve since this is in our school Maths textbook!

User 123 - 6 years ago

Ah. o.k.. Well, it's still a good problem and solution. :)

Brian Charlesworth - 6 years ago

Thank you sir! :) I would like to let you know about this question .

Rohit Ner - 6 years ago

Nice problem; I've posted a (short) solution. I've never seen the word "aam" before, so I've learned something new. :)

Brian Charlesworth - 6 years ago

@Brian Charlesworth – Just for the record, the word "aam" is actually not even written that way in Hindi. The Hindi word is actually like this: आम। The आ translates to "aa" (it has that same sound too) and the म translates to the "m" part (it actually sounds more like "muh"). The । part symbolizes full stop.

Prasun Biswas - 6 years ago

Why are we assuming that C, D, E are having the same common difference as that of A, B, C ?? Is it mentioned in the qs? Or any implicit global convention to consider two A.P.-s have same common difference when not mentioned otherwise??

Ananya Aaniya - 4 years, 7 months ago

First C,D,E do not have common difference it is their reciprocals that have common difference. Also they FORM an AP so it will be like 1,2,3 not any terms of AP. Even though if the common difference of reciprocals and the first AP is not same then also the result will be same. You can try it by taking examples.

Adwyck Gupta - 2 years, 5 months ago

Ravi Dwivedi
Jul 5, 2015

Moderator note:

Other than algebraic manipulation, is there another way to see why $a e = c ^2$ ?

For example, if a, b, c, d were an arithmetic progression, c, d, e was a geometric progression and d, e, f, g was a harmonic progression, do we have $d^2 = ce = bf = ag ?$

JEE progressions

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