# Problems involving Geometric Progressions: Difficult Problems with Solutions

Problem 1
Determine $$a_3$$, if $$a_n$$ is a geometric progression and
$$\begin{array}{|l}a_4-a_2=18\\a_5-a_3=36\end{array}$$.
Problem 2
Given a geometric progression $${a_n}$$, whose firtst term is 15 and common ratio $$r=-4$$. Find its sixth term.
Problem 3
Let $$a_n$$ be a geometric progression, defined as $$a_1=2$$(first term) and common ratio $$r=-2$$. Find the sum of its first 10 elements.
Problem 4
Find the sum of the first 5 powers of 7.
Problem 5
Let $${a_n}$$ be an alternating geometric progression. If $$a_1=5$$ and $$a_7=405$$, determine the value of $$a_4$$
Problem 6
Find the product of the first 7 terms of the geometric progression $${a_n}$$, defined as:
$$a_1=\frac{2}{11^3}$$, $$r=11$$.
Problem 7
Find the sum of the infinite geometric series $$a_n=6.(\frac{1}{3})^n$$
Problem 8
Find the sum of the infinite geometric series, explicitly defined by $$a_n=\frac{2^n}{3^{n+1}}$$
Problem 9
Find the sum of the first four terms of the geometric progression $${a_n}$$, for which $$a_n=\frac{2.3^n}{5}$$
Problem 10
Find the common ratio r of an infinite geometric series with sum $$S=7$$ and first term 4.

Problem 11
Find the common ratio of an infinite geometric series with first term 9 and sum of terms 15.
Problem 12
Determine the common ratio r of a geometric progression $$\{a_n\}$$, for which the first term $$a_1=1$$ and the sum of the first four terms is $$S_4=40$$
Problem 13
The sum an infinite geometric series is $$S_1=6$$. The sum of the squares of all terms of the same progression is $$S_2=18$$. Find the first term of the progression.
Problem 14
Find $$0.272727(27)$$ as a fraction.
Problem 15
Find the second term of a geometric progression $$\{a_n\}$$, which satisfies
$$\begin{array}{|l}a_2+a_5-a_4=10\\a_3+a_6-a_5=20\end{array}$$
Problem 16
Find the first term of a geometric progression that second term is 2 and sum to infinity is 8.
Problem 17
Let $$x_1, x_2$$ be the roots of the equation $$x^2-3x+a=0$$ and $$y_1,y_2$$ be the roots of the equation $$x^2-12x-b=0$$. If $$x_1,x_2,y_1,y_2$$ form an increasing geometric progression in this order, determine the value of $$a\cdot b$$.

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