Problems involving Geometric Progressions - Normal Problems with Solutions

Problem №1
Given a geometric progression [tex]{a_n}[/tex], for which [tex]a_1=15[/tex] and [tex]q=-4[/tex] find its sixth member.
Problem №2
Find the second member 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 №3
Find [tex]0,272727(27)[/tex] as a fraction.
Problem №4
The sum of the members of an infinite geometric progression is [tex]S_1=6[/tex]. The sum of the squares of all members of same progression is [tex]S_2=18[/tex]. Find the first member of the progression.
Problem №5
Deterine the quotient q of a geometric progression [tex]\{a_n\}[/tex], for which [tex]a_1=1[/tex] and [tex]S_4=40[/tex]
Problem №6
Find the quotient q of an infinite geometric progression [tex]\{a_n\}[/tex], for which S=15 and a_1=9
Problem №7
Find the quotient q of an infinite geometric progression [tex]\{a_n\}[/tex], for which [tex]S=7[/tex] and [tex]a_1=4[/tex]
Problem №8
Find the sum of the first four members of the geometric progression [tex]{a_n}[/tex], for which [tex]a_n=\frac{2.3^n}{5}[/tex]
Problem №9
Find the sum of the infinite geometric progression, explicitly defined by [tex]a_n=\frac{2^n}{3^{n+1}}[/tex]
Problem №10
Find the sum of the infinite geometric progression [tex]a_n=6.(\frac{1}{3})^n[/tex]
Problem №11
Find the product of the first 7 members of the geometric progression [tex]{a_n}[/tex], defined as:
[tex]a_1=\frac{2}{11^3}[/tex], [tex]q=11[/tex].
Problem №12
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 №13
Find the sum of the first 5 powers of 7.
Problem №14
Let [tex]a_n[/tex] be a geometric progression, defined as [tex]a_1=2[/tex] and [tex]q=-2[/tex]. Find the sum of its\' first 10 elements.
Problem №15
Find the first term of a geometric progression that second term is 2 and sum to infinity is 8.
Problem №16
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 said order, determine the value of [tex]a.b[/tex].
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