# 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:

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

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