If p, q, r are in AP ; ℓ, m, n are in HP ; p ℓ, qm, r n are in GP(p ℓ≠q m ≠r n), then p: q: r= (

If p, q, r are in AP ; ℓ, m, n are in HP ; p ℓ, qm, r n are...

If $\mathrm{p}, \mathrm{q}, \mathrm{r}$ are in $\mathrm{AP} ; \ell, \mathrm{m}, \mathrm{n}$ are in $\mathrm{HP} ; \mathrm{p} \ell, \mathrm{qm}, \mathrm{r} \mathrm{n}$ are in $\mathrm{GP}(\mathrm{p} \ell \neq \mathrm{q} \mathrm{m} \neq \mathrm{r} \mathrm{n})$, then $\mathrm{p}: \mathrm{q}: \mathrm{r}=$ (a) $\mathrm{m}: \mathrm{n}: \ell$ (b) $\frac{1}{n}: \frac{1}{m}: \frac{1}{\ell}$ (c) $1: 2: 3$ (d) $\mathrm{n}: \mathrm{m}: \ell$

If $\mathrm{p}, \mathrm{q}, \mathrm{r}$ are in $\mathrm{AP} ; \ell, \mathrm{m}, \mathrm{n}$ are in $\mathrm{HP} ; \mathrm{p} \ell, \mathrm{qm}, \mathrm{r} \mathrm{n}$ are in $\mathrm{GP}(\mathrm{p} \ell \neq \mathrm{q} \mathrm{m} \neq \mathrm{r} \mathrm{n})$, then $\mathrm{p}: \mathrm{q}: \mathrm{r}=$
(a) $\mathrm{m}: \mathrm{n}: \ell$
(b) $\frac{1}{n}: \frac{1}{m}: \frac{1}{\ell}$
(c) $1: 2: 3$
(d) $\mathrm{n}: \mathrm{m}: \ell$

Key Concepts

Geometric Progression

A geometric progression (GP) is a sequence where each term after the first is found by multiplying the previous term by a fixed non-zero number called the common ratio. This multiplicative relationship introduces exponential behavior in the sequence and is crucial when relating products of terms from different sequences.

Relating Different Progressions

In problems where terms from arithmetic, harmonic, and geometric progressions are interconnected, one must utilize the defining properties of each progression to set up equations. The arithmetic progression provides linear relationships, the harmonic progression provides relations through reciprocals, and the geometric progression gives multiplicative constraints. Combining these properties allows one to derive ratios and relationships among the unknown variables involved.

Arithmetic Progression

An arithmetic progression (AP) is a sequence of numbers in which the difference between any two successive terms is constant. This constant, known as the common difference, gives the sequence a linear nature, allowing relationships among the terms (for example, the middle term is the average of the first and last terms).

Harmonic Progression

A harmonic progression (HP) is a sequence of numbers whose reciprocals form an arithmetic progression. This means that while the terms themselves are not linearly spaced, their reciprocals exhibit the regularity of an arithmetic progression, and properties of means such as the harmonic mean become relevant.

Transcript

00:01 I have even cqr in ap, l -mn in hp, pl qmrn in gp, pl not equal to qm not equal to rn.

00:08 We want to find out this question.

00:10 To start with this year, we use this relation, all this variation we use and come to the conclusion of these two is to us.

00:17 Let's look at how.

00:19 So from here we have q equals p plus r.

00:25 Next from here, and p we get m equals 2ln over l plus n.

00:31 From gp's this will be q square m square equals we have p l r n just be important now what we can do we just use q from here q equals p plus r over two and from here we use m we use m and you put in this so we get p plus r over two holskill times we have m 2 l n over l plus n all square equals p r as n now just work on this relation so simplified this here there's a value of q in this and n this take a square redistributed we get this applying gear p plus r whole square l plus n whole square and so all this we get p over r plus r plus r equals l over n plus n over l this we have got here now if multiplying with p over r we get p over r whole square and next being a negative l over n plus n over l p over r and then plus 1 equal 0 now we just work on a equation the factorizes now come up to be p over r negative l over n times p over r negative n over l to 0 from here we get p over r equals l over n or p over r equals n over l this one is not not valid because p l not equal to nr it is given in the question so we go with this plus one p over r equals l over n or from here we get from uh we get p over over from here we get so we get p times n because we have l times r or we have p over n equals r over 1 over l.

02:51 Now of third relation which we got earlier, we got q square m square equals p l r n.

02:59 From there we have q square m square equals we have now...

Please give Ace some feedback