Answer:
The geometric sequence is
a
n
=
a
r
n
−
1
=
3
⋅
2
n
−
1
.
Explanation:
In a geometric sequence, the terms are separated by a common ratio
r
. So, for example, the 4th term
a
4
will be
r
×
a
3
, the 3rd term
a
3
=
r
×
a
2
, and so on. From this we can get a general formula for the
n
th
term in terms of
r
and the first term
a
1
:
a
n
=
r
×
a
n
−
1
=
r
1
a
n
−
1
a
n
=
r
×
(
r
×
a
n
−
2
)
=
r
2
a
n
−
2
a
n
=
r
×
r
×
(
r
×
a
n
−
3
)
=
r
3
a
n
−
3
a
n
=
...
a
n
=
r
n
−
1
×
a
n
−
(
n
-
1
)
=
r
n
−
1
a
1
This is often written with the initial value
a
1
(often just called
a
) in front, like this:
a
n
=
a
r
n
−
1
For the sequence
3
,
6
,
12
,
24
, we are given the first term
a
=
3
. Now all we need is the common ratio
r
. This can be found by computing the ratio of any two successive terms.
(That is, since
a
2
=
r
×
a
1
for any geometric sequence, we can find
r
by solving
r
=
a
2
a
1
, or
r
=
a
3
a
2
, or in general
r
=
a
k
a
k
−
1
.)
Using
a
2
=
6
and
a
1
=
3
, we get
r
=
a
2
a
1 The geometric sequence is
a
n
=
a
r
n
−
1
=
3
⋅
2
n
−
1
.
Explanation:
In a geometric sequence, the terms are separated by a common ratio
r
. So, for example, the 4th term
a
4
will be
r
×
a
3
, the 3rd term
a
3
=
r
×
a
2
, and so on. From this we can get a general formula for the
n
th
term in terms of
r
and the first term
a
1
:
a
n
=
r
×
a
n
−
1
=
r
1
a
n
−
1
a
n
=
r
×
(
r
×
a
n
−
2
)
=
r
2
a
n
−
2
a
n
=
r
×
r
×
(
r
×
a
n
−
3
)
=
r
3
a
n
−
3
a
n
=
...
a
n
=
r
n
−
1
×
a
n
−
(
n
-
1
)
=
r
n
−
1
a
1
This is often written with the initial value
a
1
(often just called
a
) in front, like this:
a
n
=
a
r
n
−
1
For the sequence
3
,
6
,
12
,
24
, we are given the first term
a
=
3
. Now all we need is the common ratio
r
. This can be found by computing the ratio of any two successive terms.
(That is, since
a
2
=
r
×
a
1
for any geometric sequence, we can find
r
by solving
r
=
a
2
a
1
, or
r
=
a
3
a
2
, or in general
r
=
a
k
a
k
−
1
.)
Using
a
2
=
6
and
a
1
=
3
, we get
r
=
a
2
a
1
=
6
3
=
2
.
Thus, the common ratio is 2, the first term is 3, and so the formula for this geometric sequence is
Step-by-step explanation:
Answer: B 32,769
Step-by-step explanation:
In a geometric sequence, the consecutive terms differ by a common ratio. The formula for determining the sum of n terms, Sn of a geometric sequence is expressed as
Sn = (ar^n - 1)/(r - 1)
Where
n represents the number of term in the sequence.
a represents the first term in the sequence.
r represents the common ratio.
From the information given,
a = 3
r = - 6/3 = - 2
n = 12
Therefore, the sum of the first 15 terms, S15 is
S15 = (3 × - 2^(15) - 1)/- 2 - 1
S15 = (3 × -32769)/- 3
S15 = -98307/- 3
S15 = 32769
