what is the common ratio of the geometric sequence k-3,k+2,k+3?
Answers
Answer:
Step-by-step explanation:
If
k
+
1
,
4
k
,
3
k
+
5
is a geometric sequence
then the ratio between successive terms is equal.
k
+
1
4
k
=
4
k
3
k
+
5
⇒
(
k
+
1
)
(
3
k
+
5
)
=
(
4
k
)
2
⇒
3
k
2
+
8
k
+
5
=
16
k
2
⇒
13
k
2
−
8
k
−
5
=
0
We might be able to factor this directly or we could use the quadratic formula to determine the roots:
XXX
k
=
8
±
√
(
−
8
)
2
−
4
(
13
)
(
−
5
)
2
(
13
)
XXXX
=
8
±
√
324
2
(
13
)
XXXX
=
8
±
√
324
2
(
13
)
XXX
=
8
±
18
2
(
13
)
XXXX
=
4
±
9
13
XXXX
=
13
13
=
1
or
=
−
5
13
We could (and probably should) verify these results by checking that for each of these values of
k
the given sequence is geometric.
If
k
=
1
then
k
+
1
,
4
k
,
3
k
+
5
becomes
2
,
4
,
8
with an obvious common ratio of
2
If
k
=
−
5
13
then
k
+
1
,
4
k
,
3
k
+
5
becomes (with a little more effort)
8
13
,
−
20
13
,
50
13
with a common ratio of
(
−
5
2
)
k - 3, k + 2, k + 3, ...
To find the common ratio, find the value of k.
r = T2/T1 = T3/T2
(k + 2)/(k - 3) = (k + 3)/(k + 2)
Using cross multiply
= (k + 2)(k + 2) = (k + 3)(k - 3)
k = -13/4 or -3 1/4
Then
k - 3, k + 2, k + 3, ...
-3 1/4 - 3 = -6 1/4 -T1
-3 1/4 + 2 = -1 1/4 - T1
-3 1/4 + 3 = -1/4 - T3
-6 1/4, -1 1/4, -1/4, ...
Common ratio r = (-6 1/4)/(-1 1/4) = 5
(-1 1/4)/(-1/4) = 5
Therefore, the common ratio is 5
Hope this will be helpful to you.