How many geometric progressions are there containing the terms 48,27 and 64 ? (a) 1 (b) 3 (c) 9

How many geometric progressions are there containing the...

Key Concepts

Geometric Progression

A geometric progression is a sequence of numbers in which each term after the first is obtained by multiplying the previous term by a constant factor known as the common ratio. This property allows the sequence to be defined in a compact form, and understanding the structure is essential when determining if particular numbers can be arranged in such a sequence.

Common Ratio

The common ratio is the fixed number by which each term in a geometric progression is multiplied to obtain the next term. It governs the behavior of the sequence and is crucial in determining whether a set of numbers can be organized into a geometric progression and under what conditions.

Sequence Arrangement

When considering whether specific numbers can appear in a geometric progression, the order in which the terms occur is important. The numbers may not necessarily be consecutive in the original list, and exploring different arrangements can reveal various valid geometric sequences that include the given numbers.

Term Positioning in Sequences

This concept involves analyzing the positions that specific terms can occupy within a sequence and understanding how these positions affect the common ratio and overall validity of the progression. Determining relative positions is key when a problem asks about the existence of multiple geometric progressions containing a given set of numbers.

Transcript

00:01 How many geometric portions are possible? the terms containing 48, 27 and 64.

00:06 So we can select this is the 48, it's the ph term.

00:11 This is a f u.

00:12 This is the rth term, the gp.

00:15 So we have common ratio is up.

00:20 A is the first term.

00:26 If we just work on this, we have a, r to the part p, negative 1, that equals 48.

00:33 Next, a, r to the part 2, negative 1, that equals 0 .27.

00:37 Next a r to the power r negative one that equals 64 this we get now so what you can do now just to clear this equation by this equation so you get a r to the power r negative over a r to the part p negative one that equals 24 over 48 that will be 4 over 3 so from here you get r to the part r negative p equals 4 over 3 so we go next we do this equation by this equation so we get r to the part r negative q that equals to 4 over 27 so that's going to be 4 over 3 all cube this we go if you compare this two equation this question 1 and equation 2 if you compare this two equation so we get r to the part i'm sorry r of the part r negative q small r negative q i add to the part small a negative cube that we get r to the power smaller negative q that we get equal to we will have a here we have four over three right so four over three that to the part r negative fee that will be r to the part three times r negative three that we get here just compare now we'll have r negative q plus three r i'm sorry r negative q equals we have equals three r negative 3p or from here we have two are negative 3p positive q equal to we've got an equation here with three variables three unknowns we define the number of geometric progression possible let's take any value let's take uh let's say so it's pqr if it takes here qs 5 r s 10 let me solve this we get it as uh 20 negative 3p and positive 5 equals 0.

02:58 So i get at 25 negative 3 p is equal to 0.

03:05 I'll run in this way.

03:06 But p is not coming up to be an integer.

03:09 So let's write some more values here.

03:11 Because it's coming up to be infinite number of solutions.

03:13 We have three and no and one equation that lead to infinite number of solutions...

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