Determine Whether the Statement Is True or False

Multiple Choice

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. ​
If a₁, a₂, a₃, ... , an and b₁, b₂, b₃, ... , bn are geometric progressions, then a₁ b₁, a₂ b₂, a₃ b₃, ... , an bn is also a geometric progression.
​

A) It is false.
If
A₁, a₂, a₃, a₄, ... = 1, 3, 6, 9,..
B₁, b₂, b₃, b₄, ... = 2, 4, 8, 16,..
Are geometric progressions, then we can see that
A₁ b₁, a₂ b₂, a₃ b₃, ... = 2, 12, 48, ...
Is not a geometric progression.
B) It is true.
A geometric progression is completely determined if the first term and the common ratio are known. Thus, if
A₁, a₂, a₃, ... , a_n , ...
Is a geometric progression with the first term given by a and common ratio given by r, then by definition,
A₁ = a
A₂ = a₁ r = ar
A₃ = a₂r = ar^2
.
.
.
An = an_-1 = ar ^{n - 1}
Thus, we see that the product of two nths terms of two geometric progressions is given by
A _n b _n = a r ^{n - 1} b where a,b are the first terms of these progressions and r, r₁ are their common ratios respectively. Then we obtain that
A₁, b₁, a₂, b₂, a₃, b₃, ... , an, bn is also geometric progression with first term ab and common ratio rr₁.