Arithmetic and Geometric Sequences
Term
each number in the sequence
common ratio
the product/quotient between any two consecutive terms in a geometric sequence
common difference
the sum/difference between any two consecutive terms in an arithmetic sequence
Recursive
uses previous term, when defining using this process you must define at least one previous term
aη=-6n+5
-1, -7, -13, -19 ....
gη=(-125)(1/5)ⁿ⁻¹
-125, 25, -5, 1, ...
gη=(-125)(2/5)ⁿ⁻¹
-125, 50, -10, 4, ...
aη=n-3
-2, -1, 0, 1, 2, ....
aη=-n-3
-4, -5, -6, -7....
aη=1/2n-6
-5.5, -5, -4.5, -4, .....
geometric common ratio ½
1, 2, 4, 8, ...
geometric common ratio 3
1, 3, 9, 27, , , ...
geometric common ratio 1/6
1080, 180, 30, 5, , , ....
geometric common ratio 1/4
128, 32, 8, 2, , , ...
Find the first term in a sequence where the common difference is 6 and the eleventh term is 77.
17
geometric common ratio 1/3
189, 63, 21, 7, , , ...
gη=(2)(1/2)ⁿ⁻¹
2, 1, 1/2, 1/4, ...
gη=(2)(2)ⁿ⁻¹
2, 4, 8, 16, ...
Find the 10th term in the sequence: -5, -2, 1, 4, ...
22
Find the 72nd term in the sequence: -5, -1, 3, 7, ..
279
aη=-3n+6
3, 0, -3, -6, ....
a₁=3 aη=aη-₁ - 2
3, 1, -1, -3, ...
a₁=34 aη=aη-₁ + 9
34, 43, 52, 61
geometric common ratio 5
4, 20, 100, 500, , , ...
a₁=5 aη=aη-₁ + 5
5, 10, 15, 20, ...
geometric common ratio 2
6, 12, 24, 48, , , ...
geometric common ratio 4
6, 24, 96, 384, , , ...
geometric common ratio 6
6, 36, 216, 1296, , , ... d= or r= ?
arithmetic common difference -9
62, 53, , , 26, 17, ...
arithmetic common difference -8
64, 56, , , 32, 24, ...
g₁=8 gη=(-2)gη-₁
8, -16, 32, -64, ...
g₁=8 gη=(3/2)gη-₁
8, 12, 18, 27, ....
g₁=8 gη=(1/2)gη-₁
8, 4, 2, 1, ...
arithmetic common difference -5
92, 87, 82, 77, , , ...
Find the 21st term in the sequence. -4, 1, 6, 11, 16, ...
96
arithmetic common difference 3
97, 100, 103, 106, , , ...
18,54,162
What are the three missing terms? 6__,__,__486
10,20
What are the two missing terms? 5,__,__40
3
What is the common ratio? 2, 6, 18, 54..
2
What is the common ratio? 5,__,__40
1/4
What is the common ratio? 64, 16, 4 , 1...
3
What is the common ratio? 6__,__,__486
Is the pattern an arithmetic sequence? Why? 2, 5, 8, 11, _______, _______
Yes, each number is the previous number plus 3
Arithmetic Sequence
adding or subtracting the same number
486
Find f(6) for the given sequence: 2, 6, 18...
(1/5)
Find the common ratio of the sequence: 1/4, 1/20, 1/100
3
Find the common ratio of the sequence: 4, 12, 36
(1/3)
Find the common ratio of the sequence: 81, 27, 9
4
Find the common ratio: 1, 4, 16, 64
t(n) = 48(1/2)ⁿ⁻¹
Find the equation of the sequence: 48, 24, 12, 6, 3
f(n)=1(4)ⁿ⁻¹
Find the explicit rule for the sequence: 1, 4, 16...
f(n)=125(1/5)ⁿ⁻¹
Find the explicit rule for the sequence: 125, 25, 5...
1, -4, 16, -64, 256
Find the first five terms of the sequence: t(n) = 1(-4)ⁿ⁻¹
243, 729
Find the next 2 numbers 3, 9, 27, 81
-192, 768, -3072
Find the next three terms of the sequence: 3, -12, 48
Is the pattern an arithmetic sequence? Why? 2, 4, 8, 16, _______, _______
No, each number is the previous number multiplied by 2
arithmetic common difference 7
, , 29, 36, 43, 50, ...
arithmetic common difference 6
, , 30, 36, 42, 48, ...
arithmetic common difference 4
, , 62, 66, 70, 74, ...
arithmetic common difference 5
, , 68, 73, 78, 83, ...
arithmetic common difference -4
, , 83, 79, 75, 71, ...
arithmetic common difference -7
, , 85, 78, 71, 64, ...
arithmetic common difference -2
, , 95, 93, 91, 89, ...
Geometric
Determine if the sequence is arithmetic, geometric or neither: 100, 50, 25, 12.5
1/5
Determine the common ration: 500, 100, 20
125
Find f(1) for the sequence: 125, 25, 5...
54
Find f(3) for the given sequence: 6, 18, 54...