Difficult

# Problems involving Geometric Progressions: Very Difficult Problems with Solutions

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

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