The first term of a geometric sequence is 5 and the common ratio is 2. Which expression correctly gives the 4th term?
A5 + (4 × 2) = 13
B5 × 2⁴ = 80
C5 × 2³ = 40
D5 × 4 = 20
The nth term formula is aₙ = a₁ × r^(n−1). For the 4th term: a₄ = 5 × 2^(4−1) = 5 × 2³ = 5 × 8 = 40. The exponent is n−1 (not n) because the first term requires zero multiplications by r — it already is a₁. Option B (5 × 2⁴) is the most tempting wrong answer; students who forget the −1 choose it. Option A is the arithmetic mistake: adding r × n instead of multiplying by r^(n−1).
Question 2 Multiple Choice
A sequence reads: 100, 50, 25, 12.5. A student says this is arithmetic because 'the terms get smaller by a consistent pattern.' What is wrong with this reasoning?
AThe student is right — the terms do decrease consistently, making it arithmetic
BThe differences between consecutive terms are not constant (−50, −25, −12.5), so it cannot be arithmetic; it is geometric with r = 0.5
CIt is neither arithmetic nor geometric because the terms eventually reach zero
DThe student is right, but should call it a 'decreasing' arithmetic sequence
In an arithmetic sequence, you subtract the same constant each time. Here the differences are −50, −25, −12.5 — they keep halving, so the differences are not constant. Instead, divide consecutive terms: 50/100 = 25/50 = 12.5/25 = 0.5. The ratio is constant, making this a geometric sequence with r = 0.5. This models exponential decay: each term is half the previous one. The misconception is equating 'consistent decrease' with 'arithmetic' — what matters is whether you're adding or multiplying.
Question 3 True / False
In the formula aₙ = a₁ × r^(n−1), the exponent is n rather than n−1.
TTrue
FFalse
Answer: False
The exponent must be n−1, not n. The first term (n = 1) requires zero multiplications by r — it is simply a₁ × r⁰ = a₁ × 1 = a₁. The second term requires one multiplication (r¹), the third requires two (r²), and so on. Using r^n instead would give a₁ × r for the first term, which is actually the second term. This off-by-one error is one of the most common mistakes when applying the formula.
Question 4 True / False
A geometric sequence with a negative common ratio will generally decrease toward zero.
TTrue
FFalse
Answer: False
A negative common ratio causes the sequence to alternate in sign, but does not determine whether it grows or shrinks. If |r| > 1 (e.g., r = −3), the terms grow in absolute value while flipping sign: 2, −6, 18, −54, ... If |r| < 1 (e.g., r = −0.5), the terms do shrink toward zero while alternating. The behavior depends entirely on whether |r| is greater than, equal to, or less than 1 — the sign of r only controls the alternating pattern.
Question 5 Short Answer
Why does the formula for the nth term of a geometric sequence use r^(n−1) rather than r^n?
Think about your answer, then reveal below.
Model answer: Because the first term requires zero multiplications by the common ratio. Starting from a₁, you multiply by r once to reach the second term, twice to reach the third, and so on. By the nth term, you have multiplied by r exactly n−1 times. Using r^n would overcount by one multiplication, shifting every term up by one position.
This connects directly to exponent rules: r^0 = 1, so the first term is a₁ × 1 = a₁ (unchanged). Each subsequent term adds one power of r. The formula aₙ = a₁ × r^(n−1) captures this by starting the exponent at zero and incrementing it by one for each subsequent term. Students who understand this reasoning — rather than memorizing 'subtract 1 from n' — are far less likely to make the off-by-one error.