Geometric progressiob

If the sum of four consective terms of geometric progression is 80 and arithmetic mean of second and fourth term is 30 then find terms?

Result

a =  2
b =  6
c =  18
d =  54

Solution:

a+b+c+d=80 (b+d)/2=30 b=qa c=qb d=qc  a+c+2⋅ 30=80 a+c=20  a+aq2=20  a(1+q2)=20 qa(1+q2)=60 q=60/20=3  a=201+q2=201+32=2a+b+c+d=80 \ \\ (b+d)/2=30 \ \\ b=qa \ \\ c=qb \ \\ d=qc \ \\ \ \\ a+c + 2 \cdot \ 30=80 \ \\ a+c=20 \ \\ \ \\ a + aq^2=20 \ \\ \ \\ a(1+q^2)=20 \ \\ qa(1+q^2)=60 \ \\ q=60/20=3 \ \\ \ \\ a=\dfrac{ 20 }{ 1+q^2 }=\dfrac{ 20 }{ 1+3^2 }=2
b=q⋅ a=3⋅ 2=6b=q \cdot \ a=3 \cdot \ 2=6
c=q⋅ b=3⋅ 6=18c=q \cdot \ b=3 \cdot \ 6=18
d=q⋅ c=3⋅ 18=54d=q \cdot \ c=3 \cdot \ 18=54

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