Geometric seq

Find the third member of geometric progression if a1 + a2 = 36 and a1 + a3 = 90. Calculate its quotient.

Correct result:

c1 =  81
c2 =  373248
q1 =  3
q2 =  -0.5

Solution:

a+b=36 b=qa a+c=90 c=q⋅ b a+qa=36 a+q2 a=90 a=36/(1+q) 36+q2∗36=90∗(1+q)  36+q2⋅ 36=90⋅ (1+q) 36q2−90q−54=0  a=36;b=−90;c=−54 D=b2−4ac=902−4⋅36⋅(−54)=15876 D>0  q1,2=−b±D2a=90±1587672 q1,2=90±12672 q1,2=1.25±1.75 q1=3 q2=−0.5   Factored form of the equation:  36(q−3)(q+0.5)=0 a1=36/(1+q1)=36/(1+3)=9 a2=36/(1+q2)=36/(1+(−0.5))=72 c1=a1⋅ q12=9⋅ 32=81

Our quadratic equation calculator calculates it.

c2=a2⋅ a22=72⋅ 722=373248
q1=3
q2=(−0.5)=−12=−0.5

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