Problems involving Progressions - Problems with Solutions

Problem 1
The numbers [tex]1,b,c[/tex] form an arithmetic progression and the numbers [tex]1,b,c+1[/tex] form a geometric progression. Find c.
Problem 2
The numbers [tex]a,b,c[/tex] form an arithmetic progression with sum [tex]a+b+c=341[/tex], while [tex]a-1, b+2, c+13[/tex] form a geometric progression. Find the sum of the terms of the geometric progression.
Problem 3
The sequence [tex]a,1,b[/tex] is a non-constant arithmetic progression. The sequence [tex]1,a,b[/tex] is a geometric progression. Find b.
Problem 4
Between the numbers 9 and 243 are the numbers a and b, such that [tex]9,a,b,243[/tex] is a geometric progression. Find the arithmetic mean of a and b.
Problem 5
The numbers [tex]3,9,a[/tex] form an arithmetic progression. The sequence [tex]5,a,b[/tex] is a geometric progression. Find b.
Problem 6
The numbers [tex]2,4,x[/tex] form a geometric progression and the sequence [tex]3,x,y[/tex] is an arithmetic progression. Determine the value of y.
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