Problems involving Progressions: Difficult Problems with Solutions

Problem 1
We are given the positive numbers [tex]a,b,c[/tex], which form an arithmetic progresion in the given order. We know that [tex]a+b+c=9[/tex]. The numbers [tex]a+1, b+1, c+3[/tex] form a geometric progression in the given order. Find c.
Problem 2
Between the number 3 and the number [tex]b>3[/tex] lies the number [tex]a[/tex], such that [tex]3,a,b[/tex] form an arithmetic progression. The numbers [tex]3,a-6,b[/tex] form a geometric progression. Find a.
Problem 3
An arithmetic progression [tex]\{a_n\}[/tex]has 9 elements. [tex]a_1=1[/tex] and [tex]S_a=369[/tex] (the sum of all elements is 369). A geometric progression [tex]\{b_n\}[/tex] also has 9 elements. [tex]a_1=b_1[/tex] and [tex]a_9=b_9[/tex]. Determine the value of [tex]b_7[/tex].
Problem 4
The numbers [tex]a,b,c[/tex] are all different and form an arithmetic progression in said order. The numbers [tex]b,a,c[/tex] form a geometric progression. Find the common ratio r of the geometric progression (assume that [tex]|r| > 1[/tex]).
Problem 5
The positive numbers [tex]a,b,c[/tex] form an arithmetic progression, and [tex]a+b+c=21[/tex]. If the numbers [tex]a+2, b+3, c+9[/tex] form a geometric progression, find c.
Problem 6
The numbers [tex]a,b,c,64[/tex] form a geometric progression. [tex]a,b,c[/tex] are also respectively the first, fourth and eighth members of a non-constant arithmetic progression. Determine the value of [tex]a+b-c[/tex].
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