Question

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$If \:a_{5} = x , \:a_{11} = y , \\and \:a_{17} = z\:in \:G.P\:then \\prove\:that \: y^{2} = xz$​

Answer 1

Hey there!

General formula of nth term of G.P:

$ar^{n-1}$

where,

a = first term , r = common ratio.

Given,

a₅ = x

ar⁴ = x

or, x = ar⁴.........(i)

a₁₁ = y

ar¹⁰ = y

or, y = ar¹⁰........(ii)

and, a₁₇ = z

i.e ar¹⁶ = z

or, z = ar¹⁶......(iii)

Now, Using Equations (i) , (ii) and (iii). we get:

y² = xz

(ar¹⁰)² = ar⁴ * ar¹⁶

a²r²⁰ = a²r²⁰

Since, L.H.S = R.H.S

Hence, Proved.

Answer 2

Answer:

a5 = x

a11 = y

a17 = z

These are in GP

let r be common difference

As a5 = x

so a11 = x. r^6=y As its 6 th term after 5 th term

a17 = x. r^12 =z As its 6 th term after 11 th term

y^2 = x^2 r^12

xz = x .( x r^12) = x^2. r^12

So, y^2 = xz

Hence proved

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