# Problems involving Progressions: Difficult Problems with Solutions

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

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