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?

Correct result:

a =  2
b =  6
c =  18
d =  54

Solution:

$$\begin{gathered} a+b+c+d=80 \\ (b+d)\mathrm{/}2=30 \\ b=qa \\ c=qb \\ d=qc \\ a+c+2\cdot 30=80 \\ a+c=20 \\ a+a{q}^2=20 \\ a(1+q^2)=20 \\ qa(1+q^2)=60 \\ q=60\mathrm{/}20=3 \\ a={\displaystyle \frac{20}{1+q^2}}={\displaystyle \frac{20}{1+3^2}}=2 \end{gathered}$$

$b=q\cdot a=3\cdot 2=6$

$c=q\cdot b=3\cdot 6=18$

$d=q\cdot c=3\cdot 18=54$

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