Question 1
A geometric sequence has first term 32 and common ratio 1/2.
What is the 4th term?
- 16
- 6
- 8
- 4
- 7
Show the worked solution
- aₙ = a × rⁿ⁻¹ = 32 × (1/2)³ = 4.
- Answer: 4.
Sequences
Arithmetic and geometric sequences — find the rule, then jump
Arithmetic and geometric sequences differ in how each term is produced. An arithmetic sequence adds a fixed common difference, so the nth term is a plus (n minus 1) times d. A geometric sequence multiplies by a fixed common ratio, so the nth term is a times r to the power (n minus 1). Identifying which one you have is the first move.
The formulas are short and the arithmetic is light, so the time cost here is almost entirely in listing terms out one by one. Jumping straight to the 20th term with the formula takes one line; writing out twenty terms takes most of a minute and introduces a chance to slip at every step.
Try it against the clock. Sequences (AP & GP) runs at 90s easy, 105s medium and 120s hard. The clock is stamped and judged on our server, so the limit you see is the deadline that is actually enforced — and it starts when you tap Start, not while the question is loading.
How to work arithmetic and geometric sequences with the nth-term formula
-
Decide whether terms are added to or multiplied
A constant gap between consecutive terms means arithmetic; a constant multiplier means geometric. Checking two gaps settles it immediately.
-
Apply the arithmetic nth term
The nth term is a + (n − 1)d. With a first term of 5 and a difference of 3, the 20th term is 5 + 19 × 3, which is 62.
-
Apply the geometric nth term
The nth term is a × r^(n−1). With a first term of 3 and a ratio of 2, the 5th term is 3 × 2⁴, which is 48.
-
Count the gaps, not the terms
Reaching the nth term takes n − 1 steps from the first. Using n instead is the single most common error in this topic.
The two sequence types side by side
| The sequence | How it works |
|---|---|
| Arithmetic, each step | Add the common difference d |
| Arithmetic, nth term | a + (n − 1)d |
| Geometric, each step | Multiply by the common ratio r |
| Geometric, nth term | a × r^(n−1) |
| First term 3, 5th term | Arithmetic (d = 2): 11 · Geometric (r = 2): 48 |
| Ratio below 1 | Geometric only — terms shrink, e.g. 16, 8, 4 |
3 sample arithmetic and geometric sequences, with worked solutions
Generated by the same engine that mints the Quant Daily daily. Each answer is computed from the numbers printed in the question, and the walkthrough below each one is the engine's own working — not a solution written afterwards. Reload this page's live drill and you get different numbers.
Question 2
A geometric sequence has first term 2 and common ratio 2.
What is the 4th term?
- 16
- 18
- 8
- 56
- 30
Show the worked solution
- aₙ = a × rⁿ⁻¹ = 2 × 2³ = 16.
- Answer: 16.
Question 3
An arithmetic sequence has first term 4 and common difference 2.
What is the sum of the first 8 terms?
- 88
- 18
- 32
- 79
- 96
Show the worked solution
- Sₙ = n/2 × (2a + (n − 1)d) = 8/2 × (8 + 7 × 2) = 88.
- Answer: 88.
Common mistakes this drill is built from
The wrong options are not random numbers. Each one is the result of a specific careless error for this topic, so picking one tells you which habit is costing you marks.
Using n instead of n − 1
There are n − 1 gaps between the first and the nth term. Using n adds one extra step and lands exactly one term too far along.
Treating a geometric sequence as arithmetic
16, 8, 4 has no common difference at all. Check whether consecutive terms share a ratio before assuming a constant gap.
Listing every term
The formula reaches the 20th term in one line. Listing works but burns the time these questions are designed to test.
Why your answer came out wrong
The specific wrong numbers this topic produces, and what each one tells you about the step you took. If you have just got a question wrong and want to know which habit did it, start here.
First term 5, common difference 3 — is the 20th term 65?
It is 62. There are 19 gaps between the first and twentieth terms, not 20, so the calculation is 5 + 19 × 3. Multiplying by n rather than n − 1 gives 65, which is the 21st term and is reliably printed as an option.
For a first term of 3 and ratio 2, is the 5th term 3 × 2⁵?
The exponent is n − 1, so it is 3 × 2⁴ = 48. Using 2⁵ gives 96, which is the sixth term. Reaching the fifth term requires four multiplications from the first, which is exactly what the n − 1 encodes.
FAQ
How do I tell an arithmetic sequence from a geometric one?
Compare consecutive terms. If you get from one to the next by adding the same amount each time, it is arithmetic; if you get there by multiplying by the same factor, it is geometric. Checking two consecutive gaps is enough to identify which formula applies.
What is the nth term of a geometric sequence?
It is a × r^(n−1), where a is the first term and r the common ratio. For a first term of 3 and a ratio of 2, the fifth term is 3 × 2⁴ = 48. The exponent is n − 1 because reaching the fifth term takes four multiplications, not five.
Can the common ratio be a fraction?
Yes, and the formula is unchanged. A sequence starting at 16 with a ratio of one half runs 16, 8, 4, 2, so the third term is 16 × (1/2)² = 4. A ratio below one makes the terms shrink rather than grow, but nothing else about the method differs.
Practise the next topic
- All 51 drill types — the shared daily cycles 36 of 51, with every published time limit.
- Function notation problems — substitute — function notation problems, timed.
- Exponent rules practice — evaluate powers and roots on sight — exponent rules practice, timed.
- Speed tricks — the seven mental-math shortcuts that make a 15-second answer possible.
- How to improve calculation speed — the six-week arc these drills fit into.
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