a The given rule means that, after the first term of the sequence, every next term a_(n+1) is the opposite of the current term a_n.
B
b The given rule means that, after the first term of the sequence, every term a_n is the difference between half the previous term a_(n-1) and 5.
C
c The given rule means that, after the first term of the sequence, every next term a_(n+1) is equal to the previous term a_(n-1).
A
aFirst five terms: 17, -17, 17, -17, and 17
Is sequence arithmetic, geometric, or neither? Sequence is geometric.
B
bFirst five terms: 32, 11, 0.5, -4.75, and -7.375
Is sequence arithmetic, geometric, or neither? Neither.
C
cFirst five terms: 81, 81, 81, 81, and 81
Is sequence arithmetic, geometric, or neither? Both arithmetic and geometric.
Practice makes perfect
a We are asked to write the first 5 terms of a sequence, given a recursive rule and identify the sequence as arithmetic, geometric, or neither. Let's start by finding the first five terms.
First Five Terms
We are given the following recursive rule for our sequence.
a_1&=17
a_(n+1)&= - a_n, for n ≥ 1To do so, we will use a table.
n
a_(n+1) = - a_n
- a_(n-1)
a_n
-
-
-
17
1
a_(1+1)=- a_1
- a_1=-( 17)
-17
2
a_(2+1)=- a_2
- a_2=-( -17)
17
3
a_(3+1)=- a_3
- a_3=-(17)
-17
4
a_(4+1)=- a_4
-a_4=-(-17)
17
Therefore, the first 5 terms of the sequence are 17, -17, 17, -17, and 17. Now, let's identify whether the sequence is arithmetic, geometric, or neither.
Is Sequence Arithmetic?
When a sequence is arithmetic, the difference between two consecutive terms is constant. Examining our sequence, we can see that this is not the case.
17 -34 → -17 +34 → 17 -34 → -17 +34 → 17
Therefore, the sequence is not arithmetic. Next, let's check whether the sequence is geometric.
Is Sequence Geometric?
When a sequence is geometric, the ratio between two consecutive terms is constant. Examining our sequence, we can see that this is the case.
17 *(-1) → -17 *(-1) → 17 *(-1) → -17 *(-1) → 17
Therefore, the sequence is geometric and the common ratio is -1.
b We are asked to write the first 5 terms of a sequence, given a recursive rule and identify the sequence as arithmetic, geometric, or neither. Let's start by finding the first five terms.
First Five Terms
We are given the following recursive rule for our sequence.
a_1&=32
a_(n+1)&= -5 + 12 a_n, for n ≥ 1To do so, we will use a table.
To do so, we will use a table.
n
a_(n+1) = -5 + 12 a_n
-5 + 12 a_n
a_n
-
-
-
32
1
a_(1+1)=-5 + 12 a_1
-5 + 12 a_1=-5 + 12( 32)
11
2
a_(2+1)=-5 + 12 a_2
-5 + 12 a_2=-5 + 12( 11)
0.5
3
a_(3+1)=-5 + 12 a_3
-5 + 12 a_3=-5 + 12(0.5)
-4.75
4
a_(4+1)=-5 + 12 a_4
-5 + 12 a_4=-5 + 12(-4.75)
-7.375
Therefore, the first 5 terms of the sequence are 32, 11, 0.5, -4.75, and -7.375. Now, let's identify whether the sequence is arithmetic, geometric, or neither.
Is Sequence Arithmetic?
When a sequence is arithmetic, the difference between two consecutive terms is constant. Examining our sequence, we can see that this is not the case.
32 -21 → 11 -10.5 → 0.5 -5.25 → -4.75 -2.625 → -7.375
Therefore, the sequence is not arithmetic. Next, let's check whether the sequence is geometric.
Is Sequence Geometric?
When a sequence is geometric, the ratio between two consecutive terms is constant. Examining our sequence, we can see that again this is not the case.
32 ≈ *0.34 → 11 ≈ *0.05 → 0.5 * (-9.5) → -4.75 * 1.55 → -7.375
Therefore, the sequence is geometric arithmetic. Therefore, our sequence is neither arithmetic nor geometric.
c We are asked to write the first 5 terms of a sequence, given a recursive rule and identify the sequence as arithmetic, geometric, or neither. Let's start by finding the first five terms.
First Five Terms
We are given the following recursive rule for our sequence.
a_1&=81
a_(n+1)&= a_n, for n ≥ 1To do so, we will use a table.
To do so, we will use a table.
n
a_(n+1) = a_n
a_n
a_n
-
-
-
81
1
a_(1+1)=a_1
a_1= 81
81
2
a_(2+1)=a_2
a_2= 81
81
3
a_(3+1)=a_3
a_3=81
81
4
a_(4+1)=a_4
a_4=81
81
Therefore, the first 5 terms of the sequence are 81, 81, 81, 81, and 81. Now, let's identify whether the sequence is arithmetic, geometric, or neither.
Is Sequence Arithmetic?
When a sequence is arithmetic, the difference between two consecutive terms is constant. Examining our sequence, we can see that this is not the case.
81 +0 → 81 +0 → 81 +0 → 81 +0 → 81
Therefore, the sequence is not arithmetic. However, we are not done yet, as the sequence might also be geometric!
Is Sequence Geometric?
When a sequence is geometric, the ratio between two consecutive terms is constant. Examining our sequence, we can see that again this is not the case.
81 *1 → 81 *1 → 81 *1 → 81 *1 → 81
Therefore, the sequence is geometric arithmetic. Therefore, our sequence is neither arithmetic nor geometric.
