Problems involving Geometric Progressions - Normal Problems with Solutions

Problem 1
Determine [tex]a_3[/tex], if [tex]a_n[/tex] is a geometric progression and
[tex]\begin{array}{|l}a_4-a_2=18\\a_5-a_3=36\end{array}[/tex].
Problem 2
Given a geometric progression [tex]{a_n}[/tex], whose firtst term is 15 and common ratio [tex]r=-4[/tex]. Find its sixth term.
Problem 3
Find the second term of a geometric progression [tex]\{a_n\}[/tex], which satisfies
[tex]\begin{array}{|l}a_2+a_5-a_4=10\\a_3+a_6-a_5=20\end{array}[/tex]
Problem 4
Find [tex]0.272727(27)[/tex] as a fraction.
Problem 5
The sum an infinite geometric series is [tex]S_1=6[/tex]. The sum of the squares of all terms of the same progression is [tex]S_2=18[/tex]. Find the first term of the progression.
Problem 6
Determine the common ratio r of a geometric progression [tex]\{a_n\}[/tex], for which the first term [tex]a_1=1[/tex] and the sum of the first four terms is [tex]S_4=40[/tex]
Problem 7
Find the common ratio of an infinite geometric series with first term 9 and sum of terms 15.
Problem 8
Find the common ratio r of an infinite geometric series with sum [tex]S=7[/tex] and first term 4.
Problem 9
Find the sum of the first four terms of the geometric progression [tex]{a_n}[/tex], for which [tex]a_n=\frac{2.3^n}{5}[/tex]
Problem 10
Find the sum of the infinite geometric series, explicitly defined by [tex]a_n=\frac{2^n}{3^{n+1}}[/tex]

Problem 11
Find the sum of the infinite geometric series [tex]a_n=6.(\frac{1}{3})^n[/tex]
Problem 12
Find the product of the first 7 terms of the geometric progression [tex]{a_n}[/tex], defined as:
[tex]a_1=\frac{2}{11^3}[/tex], [tex]r=11[/tex].
Problem 13
Let [tex]{a_n}[/tex] be an alternating geometric progression. If [tex]a_1=5[/tex] and [tex]a_7=405[/tex], determine the value of [tex]a_4[/tex]
Problem 14
Find the sum of the first 5 powers of 7.
Problem 15
Let [tex]a_n[/tex] be a geometric progression, defined as [tex]a_1=2[/tex](first term) and common ratio [tex]r=-2[/tex]. Find the sum of its first 10 elements.
Problem 16
Find the first term of a geometric progression that second term is 2 and sum to infinity is 8.
Problem 17
Let [tex]x_1, x_2[/tex] be the roots of the equation [tex]x^2-3x+a=0[/tex] and [tex]y_1,y_2[/tex] be the roots of the equation [tex]x^2-12x-b=0[/tex]. If [tex]x_1,x_2,y_1,y_2[/tex] form an increasing geometric progression in this order, determine the value of [tex]a\cdot b[/tex].
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