Statement 1 If a^2+2 bc, b^2+2 ac, c^2+2 ab are in AP then, b-c2 c-a, a-b are in HP and Statemen

step 1 Hi, statement 2 let's work on. If we have A1, A and so on A and in AP, then in that case we will

Statement 1 If a^2+2 bc, b^2+2 ac, c^2+2 ab are in AP then,...

Statement 1 If $\mathrm{a}^{2}+2 \mathrm{bc}, \mathrm{b}^{2}+2 \mathrm{ac}, \mathrm{c}^{2}+2 \mathrm{ab}$ are in AP then, $\mathrm{b}-c_{2} \mathrm{c}-\mathrm{a}, \mathrm{a}-\mathrm{b}$ are in $\mathrm{HP}$ and Statement 2 If $a_{1}, a_{2}, \ldots . a_{n}$ are in AP then $\frac{a_{1}+k}{h}, \frac{a_{2}+k}{h}, \ldots \ldots .$ in AP

Statement 1 If $\mathrm{a}^{2}+2 \mathrm{bc}, \mathrm{b}^{2}+2 \mathrm{ac}, \mathrm{c}^{2}+2 \mathrm{ab}$ are in AP then, $\mathrm{b}-c_{2} \mathrm{c}-\mathrm{a}, \mathrm{a}-\mathrm{b}$ are in $\mathrm{HP}$
and
Statement 2 If $a_{1}, a_{2}, \ldots . a_{n}$ are in AP then $\frac{a_{1}+k}{h}, \frac{a_{2}+k}{h}, \ldots \ldots .$ in AP

Key Concepts

Sequence Transformation and Invariance

Sequence transformation and invariance involves operations that modify the terms of a sequence without altering its fundamental progression type. For example, adding a constant or scaling by a non-zero factor maintains an arithmetic progression. This concept is pivotal when solving problems where sequences undergo linear transformations, ensuring that their core properties remain intact and can be analyzed effectively.

Harmonic Progression (HP)

A harmonic progression is a sequence of numbers whose reciprocals form an arithmetic progression. This relationship bridges the gap between the non-linear behavior of HPs and the linear properties of APs, making it a crucial concept in both algebra and analysis. Analyzing HPs often involves transforming them into their reciprocal AP form, which simplifies understanding their properties and behavior.

Arithmetic Progression (AP)

An arithmetic progression is a sequence where the difference between consecutive terms is constant. This regular spacing allows for easy analysis and manipulation of the sequence, as operations like addition or multiplication by a constant preserve the structure of the progression. Understanding APs is essential in many areas of algebra and number theory, particularly when studying sequential patterns and series.

Transcript

00:01 Hi, statement 2 let's work on.

00:04 If we have a1, a and so on a and in ap, then in that case we will have a1 plus k, a2 plus k, and so on an plus k.

00:15 That will be in ap, right? if we add a constant to each term, it will remain in ap.

00:20 You divide by again constant to each term.

00:23 It will also be in ap.

00:25 So statement 2 is correct.

00:27 Work on our statement 1, start with a square plus 2 basis.

00:32 Statement one we have a square plus 2bc then b square plus 2 ac then c square plus 2 a b then c square plus 2 a b square plus 2 a b square plus 2 a b square plus 2 ab negative k that's in ap right so let's know why is this want to be in a t so we can say now a square plus 2 bc negative k that's in a where here k is ab plus b plus b so that we're following from statement more following more statement so you can write it as as a s to have plus 2b negative ab negative b c negative b c then b square plus 2ac negative ab negative bc negative b c then b square plus 2 a b negative b negative b negative b negative square work on this so it's coming up to be a square negative abe plus bc negative negative c next would be b square positive asp negative ab negative bc next is coming up to be in ap c square then plus a b negative bc negative c that would be ap we can just read in the terms and we can write it as so we spectra out here we get it as a negative b c negative a we expect out that so it will be negative a negative b c negative a you can check that you get negative a c then a square then b c then negative b then negative b this will be given as negative b negative b negative c a negative b and this is given as negative c negative a and b negative state...

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