[tex] \red {14}[/tex]
[tex] If \:a_{5} = x , \:a_{11} = y , \\and \:a_{17} = z\:in \:G.P\:then \\prove\:that \: y^{2} = xz [/tex]
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[tex] \red {14}[/tex]
[tex] If \:a_{5} = x , \:a_{11} = y , \\and \:a_{17} = z\:in \:G.P\:then \\prove\:that \: y^{2} = xz [/tex]
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Hey there!
General formula of nth term of G.P:
[tex]ar^{n-1}[/tex]
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:
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
#answerwithquality #BAL