If a^2, b^2, c^2 are in AP, then (a)/(b+c), (b)/(c+a), (c)/(b+a) are in (a) AP (b) GP (c) HP (d)

If a^2, b^2, c^2 are in AP, then (a)/(b+c), (b)/(c+a),...

Key Concepts

Arithmetic Progression (AP)

An arithmetic progression is a sequence of numbers in which the difference between consecutive terms is constant. This property provides a systematic way to relate terms by addition or subtraction, which is particularly useful in problems involving the conditions that connect squares or other functions of variables.

Geometric Progression (GP)

A geometric progression is a sequence where each term after the first is obtained by multiplying the previous term by a constant factor. Unlike an arithmetic progression, the relationship here involves multiplication rather than addition, and recognizing these multiplicative relationships is key when investigating ratios or products of terms in algebraic expressions.

Harmonic Progression (HP)

A harmonic progression is a sequence of numbers whose reciprocals form an arithmetic progression. This concept is important when sequences are defined by fractions, especially when the condition in the problem leads to patterns that can be more easily identified by examining the reciprocal sequence rather than the sequence itself.

Transcript

00:01 I estimate that is square b square c square we have with an ap we're to find out for a over b plus c b over c plus a c over b plus a just work on this so we get here b square negative a square equals c square negative b square the common difference of ap we have b negative a b b positive a equals c negative b times c positive b we have now let's work on this here b over c plus a negative a over b plus c that's coming out to be b b plus c negative a a plus c over c plus a b plus c that would be we get here b square negative a square positive b c negative a c over c positive a b positive a b we've got it we're going to working on this, so we get from here to be b negative a, b positive a, from this first two terms, positive c, b negative a, over c positive a, b positive a, b positive c.

01:32 We've been working on this, we get c negative b.

01:37 I'm sorry, we get first we get here b, b negative a times a plus b plus c over c plus a, b plus c, right? now b negative a is here negative b ap common difference c negative b a plus b plus c over c plus a plus a plus b now next we get reviews on the previous equations that is given as b negative a then b positive a equals c negative b c positive so from here find out we have c negative b over a positive b b over a positive that equals b negative a over b positive c so in place of this here we just put this values here that's coming out to be equal to uh we get it as a plus b plus c time c negative b over a positive b that's given as b negative a over b positive c so let us c negative b over a positive c so let us c negative b over a positive that is given as, let me just work on this.

03:20 Or let me get simplified that is not required.

03:22 We just try to just simplify this here we have.

03:24 We simplify the numerator here.

03:26 This comes out of this cancels out.

03:28 And we get it as equals c squared, negative b square, then positive a c, negative a, c, negative a b over c plus a, a plus b.

03:45 Now that can be given as c over a plus b...

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