A sequence is an ordered list of numbers following a rule. In an arithmetic sequence, each term increases (or decreases) by a constant amount, called the common difference. In a geometric sequence, each term is multiplied by a constant amount, called the common ratio. A series is the sum of the terms in a sequence, and both types have formulas for finding a specific term or a total sum without listing every term.
Example
Weekly savings of $20, $40, $60, $80... is arithmetic with a common difference of 20, while a $100 investment growing at 10% each year (100, 110, 121, 133.10...) is geometric with a common ratio of 1.1 — very different long-term outcomes despite both "increasing."
Key terms
Arithmetic sequence:
A sequence where each term increases or decreases by a constant amount.
Geometric sequence:
A sequence where each term is multiplied by a constant ratio.
Series:
The sum of the terms in a sequence.
Questions
1. An arithmetic sequence changes by:
A constant amount added or subtracted each term
A constant amount multiplied each term
A completely random amount each term
No change at all between terms
2. A geometric sequence changes by:
A constant ratio multiplied each term
A constant amount added each term
No change at all between terms
A completely random amount each term
3. A series is:
The sum of the terms in a sequence
A single term only
A type of graph
The difference between two terms
4. The sequence 5, 10, 15, 20 is:
Arithmetic
Geometric
Neither arithmetic nor geometric
Not a sequence at all
5. The sequence 2, 4, 8, 16 is:
Geometric
Arithmetic
Neither arithmetic nor geometric
Not a sequence at all
6. The common difference in an arithmetic sequence is the amount:
Added or subtracted between consecutive terms
Multiplied between consecutive terms
Divided at the very end only
Squared at the very end only
7. The common ratio in a geometric sequence is the amount:
Multiplied between consecutive terms
Added between consecutive terms
Subtracted at the very end only
Never used in geometric sequences
8. In the arithmetic sequence 20, 40, 60, 80, the common difference is:
20
2
40
10
9. In the geometric sequence 100, 110, 121, 133.10, the common ratio is:
1.1
10
0.1
11
10. The next term in the arithmetic sequence 3, 7, 11, 15 is:
19
17
20
18
11. The next term in the geometric sequence 3, 6, 12, 24 is:
48
36
30
27
12. Weekly savings that add a fixed $20 each week is an example of:
An arithmetic sequence
A geometric sequence
Neither type of sequence
A type of series only, with no sequence
13. An investment growing by a fixed percentage each year is an example of:
A geometric sequence
An arithmetic sequence
Neither type of sequence
A type of series only, with no sequence
14. The 5th term of the arithmetic sequence starting at 4 with common difference 3 is:
16
13
19
15
15. Why do geometric sequences eventually grow (or shrink) much faster than arithmetic sequences with a similar starting rate?
Multiplying repeatedly compounds the effect, unlike consistently adding the same fixed amount
Arithmetic sequences always eventually grow faster than geometric ones
Geometric and arithmetic sequences always produce identical long-term results
Multiplication and addition produce mathematically identical growth patterns
16. Why might a fixed weekly savings plan (arithmetic) be more predictable than an investment with compounding returns (geometric), even if the geometric option grows faster overall?
Arithmetic growth adds the same fixed amount every time, while geometric growth depends on a percentage of a changing amount
Both types of growth are always equally predictable in every situation
Geometric sequences never involve any variation in value over time
Arithmetic sequences are always the faster-growing option long-term
17. If the first term is 5 and the common ratio is 2, what is the 4th term of the geometric sequence?
40
20
10
35
18. Why might understanding series formulas (for summing many terms) be more efficient than adding terms one by one?
A formula can calculate a total sum directly, without manually listing and adding potentially hundreds of terms
Series formulas are always slower than manual addition
Series formulas can only ever be used for very short sequences
Manual addition and formula-based summation always take exactly the same time
19. If the first term is 8 and the common difference is -3, what is the 6th term of the arithmetic sequence?
-7
-3
5
2
20. Why might a loan with compounding interest (geometric growth in the amount owed) become much harder to pay off the longer it's left unpaid?
Interest compounds on both the original amount and previously accumulated interest, accelerating the total owed
Compounding interest always adds the exact same fixed dollar amount each period
Loan balances only ever grow arithmetically, never geometrically
Leaving a loan unpaid for longer never has any effect on the total amount owed
21. Why might recognising whether a real-world pattern is arithmetic or geometric matter before choosing a formula to solve it?
Using the wrong formula for the pattern type will produce an incorrect result, since the two types grow in fundamentally different ways
Arithmetic and geometric formulas always produce identical results regardless of the pattern
The type of pattern never affects which formula should be applied
Formulas for sequences are always interchangeable regardless of pattern type
Answer key (parent copy)
1. A constant amount added or subtracted each term
2. A constant ratio multiplied each term
3. The sum of the terms in a sequence
4. Arithmetic
5. Geometric
6. Added or subtracted between consecutive terms
7. Multiplied between consecutive terms
8. 20
9. 1.1
10. 19
11. 48
12. An arithmetic sequence
13. A geometric sequence
14. 16
15. Multiplying repeatedly compounds the effect, unlike consistently adding the same fixed amount
16. Arithmetic growth adds the same fixed amount every time, while geometric growth depends on a percentage of a changing amount
17. 40
18. A formula can calculate a total sum directly, without manually listing and adding potentially hundreds of terms
19. -7
20. Interest compounds on both the original amount and previously accumulated interest, accelerating the total owed
21. Using the wrong formula for the pattern type will produce an incorrect result, since the two types grow in fundamentally different ways