1. Which sequence is geometric?
- 2, 4, 6, 8
- 1, 4, 9, 16
- 2, 6, 18, 54
Check the answer and reasoning
2, 6, 18, 54
Each term is multiplied by the same ratio 3.
MATH / AS level
Series and binomial expansion
Arithmetic progressions add a fixed difference; geometric progressions multiply by a fixed ratio. Summation expresses the total rather than a single term.
Learning objectives
- Explain series and binomial expansion using a concrete example.
- Apply the worked method and justify each step.
- Recognise the limitation described in the final check.
The big idea
Arithmetic progressions add a fixed difference; geometric progressions multiply by a fixed ratio. Summation expresses the total rather than a single term.
A geometric sum to infinity exists only when |r| < 1. Binomial coefficients count selections and generate polynomial expansions for nonnegative integer powers.
Equations and relationships
Infinite geometric sum.
Worked example
Find 3 + 1.5 + 0.75 + … to infinity.
Follow the solution
The first term is 3 and r=0.5, so S∞=3/(1−0.5)=6.
Investigate and explain
Predict the worked result before calculating. Change one input, record the new result, and explain which assumptions still hold.
Use the prediction activity below; a local model appears only where it supports this topic.
Check your understanding
2. Find 3 + 1.5 + 0.75 + … to infinity.
- 6
- 4.5
- Infinity
Check the answer and reasoning
6
The first term is 3 and r=0.5, so S∞=3/(1−0.5)=6.
3. Does 1 + 2 + 4 + … have a finite geometric sum?
- Yes, −1
- No
- Yes, 2
Check the answer and reasoning
No
The ratio has magnitude greater than 1; the infinite-sum formula does not apply.