Problems involving Progressions: Very Difficult Problems with Solutions

Problem 1
The numbers [tex]a,b,c[/tex], in the given order, form a non-constant geometric progression. The numbers [tex]a,2b,3c[/tex] form an arithmetic progression in the given order. Find the common ratio r of the geometric progression.
Problem 2
Let [tex]a,b,c,d[/tex] be non-integer real numbers. The numbers [tex]a,b,c[/tex] in said order form an arithmetic progression. The numbers [tex]b,c,d[/tex] in said order form a geometric progression. If [tex]a+d=37[/tex] and [tex]b+c=36[/tex], find d.
Problem 3
The integer numbers [tex]a,b,c[/tex] form a geometric progression. [tex]a,b,c-64[/tex] form an arithmetic progression. [tex]a,b-8,c-64[/tex] form a geometric progression again. Find c.
Problem 4 sent by Rohit Dalal Ajanta Public School
Find sum of the first 20 terms of [tex]3 \times 5+6 \times 10+9\times 15...[/tex]
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