Problems involving Geometric Progressions: Very Difficult Problems with Solutions

Problem 1
Let [tex]{a_n}[/tex] be a sequence of numbers, which is defined by the recurrence relation [tex]a_1=1; \frac{a_{n+1}}{a_n}=2^n[/tex]. Find [tex]log_2(a_{100})[/tex].
Problem 2
Given the sequence, defined as [tex]a_1=1; a_{n+1}-a_n=3^n[/tex], find the value of [tex]a_{10}[/tex].
Problem 3
Given the linear system [tex]\begin{array}{|l}x+y+z=a+4\\2x-y+2z=2a+2\\3x+2y-3z=1-2a \end{array} [/tex], where [tex]x,y,z[/tex] in this order form a geometric progression, find the value of the positive real parameter a.
Problem 4
[tex]a,b,c[/tex] is a geometric progression (a,b,c - real numbers). If [tex]a+b+c=26[/tex] and [tex]a^2+b^2+c^2=364[/tex], find b.
Problem 5
Find the infinite sum [tex]S=1+2.\frac{1}{7}+3.(\frac{1}{7})^2+...+(n+1).(\frac{1}{7})^n+...[/tex]
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