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.

Question

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, ... , an , ...
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 = ar2

.
.
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.

Answer 1

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, ... , an , ...
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 = ar2

.
.
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.