Investigation of Sums and Products of Arithmetic and Geometric
Sequences
Type I
1. In each case, identify the following sequences as either arithmetic of geometric and
state the common difference or common ratio as appropriate.
(a) S1 =1.3, 1.7, 2.1, 2.5, …
(b) S 2 =1.3, 1.69, 2.197, 2.8561, …
(c) S 3 =1.3, 1.27, 1.24, 1.21…
(d) S 4 =1.3, -1.625, 2.03125, -2.5390625…
2. Sequence T1 is obtained by adding the corresponding terms of S1 and S 2 so that
T1 =2.6, 3.39, 4.297, 5.3561, …. We shall write T1 = S1 + S 2 to convey the addition
of S1 and S 2 in this way. Is T1 an arithmetic or Geometric sequence?
3. Let T2 = S1 + S 3 , T3 = S 2 + S 3 , T4 = S 2 + S 4 . Investigate whether T2 , T3 and T4 are
arithmetic or geometric sequences. Whenever an arithmetic sequence is identified
state its common difference. Whenever a geometric sequence I identified state its
common ratio.
4. Sequence U 1 is obtained by multiplying the corresponding terms of S1 and S 2 so that
U 1 =1.69, 2.873, 4.69137, 7.14025, …. Is U 1 an arithmetic or Geometric
sequence?
5. Let U 2 = S1  S 3 , U 3 = S 2  S 3 , U 4 = S 2  S 4 . Investigate whether U 2 , U 3 and U 4
are arithmetic or geometric sequences. Whenever an arithmetic sequence is
identified state its common difference. Whenever a geometric sequence I identified
state its common ratio.
6. A   a1 , a1  d1 , a1  2d1 , …, B   a 2 , a 2  d 2 , a 2  2d 2 , …. Investigate
whether A  B and A  B are arithmetic or geometric sequences. If an arithmetic
sequence is identified state its common difference. If a geometric sequence is
identified, state its common ratio.
2
2
7. C   a1 , a1 r1 , a1 r1 , …, D   a 2 , a 2 r2 , a 2 r2 , …. Investigate whether C  D
and C  D are arithmetic or geometric sequences. If an arithmetic sequence is
identified state its common difference. If a geometric sequence is identified, state its
common ratio.
8. E   a1 , a1 r , a1 r 2 , …, D   a 2 , a 2 r , a 2 r 2 , …. Investigate whether E  F and
E  F are arithmetic or geometric sequences. If an arithmetic sequence is identified
state its common difference. If a geometric sequence is identified, state its common
ratio.
Please remember, the investigation on this page is only gathering the necessary data in order
for you to create, explain, and justify a hypothesis. The hypothesis is the purpose of the
paper, not the investigation.