Sequences Quiz — Arithmetic & Geometric

Q01 — Recursive generation

A sequence is defined recursively: t₁ = 5, tₙ = tₙ₋₁ + 4.
What is the value of t₅?

Start with t₁ = 5 and apply the rule tₙ = tₙ₋₁ + 4 repeatedly until you reach the 5th term.

Q02 — Common difference

The sequence 3, 7, 11, 15, … is arithmetic.
What is the common difference d?

The common difference is found by subtracting any term from the term that follows it.

Q03 — nth-term rule

Using the formula tₙ = t₁ + (n − 1)d, find the 20th term of the sequence with t₁ = 6 and d = 3.

Substitute t₁ = 6, d = 3, and n = 20 into the formula tₙ = t₁ + (n − 1)d.

Q04 — Find the first term

The 10th term of an arithmetic sequence is 47 and the common difference is d = 5.
What is the first term t₁?

Substitute t₁₀ = 47, n = 10, and d = 5 into tₙ = t₁ + (n − 1)d, then solve for t₁.

Q05 — Graphical / linear growth

An arithmetic sequence has t₁ = 10 and d = 7.
On a graph of term value vs. term number, what is the slope of the straight line through the points?

Think about what controls how steeply the points rise on the graph. What does d represent in the context of rise over run?

Q06 — Simple interest (investment)

Mia invests $2 000 at simple interest of 4% per annum.
The balance at the end of each year forms an arithmetic sequence.
What is the balance (in dollars) at the end of year 8?

First calculate the annual interest amount (4% of $2 000). This is your common difference d. Then use the nth-term formula with n = 8, adding interest to the original principal.

Q07 — Taxi fare

A taxi charges a flag-fall of $3.50 and then $2.20 per kilometre.
The total fare forms an arithmetic sequence with the distance as the term number.
What is the fare (in dollars, rounded to 2 decimal places) for a 12 km trip?

The flag-fall is your starting value and the per-km charge is the common difference. Use the arithmetic sequence structure: total fare = flag-fall + (km × rate per km).

Q08 — Straight-line depreciation

A machine is bought for $15 000 and depreciates by $1 200 per year (straight-line method).
After how many complete years will its value first fall below $9 000?

Set up the inequality: 15 000 − 1 200n < 9 000 and solve for n. Remember n must be a whole number of complete years.

Q09 — Recursive rule from sequence

The sequence is: 50, 44, 38, 32, …
Write the recursive rule as tₙ = tₙ₋₁ + d. What is d?

Subtract any term from the term that follows it. Is the sequence increasing or decreasing? That will tell you the sign of d.

Q10 — Linear decay model

A water tank holds 5 000 litres. Water drains at 150 L per hour.
How many litres remain after 18 hours?

This is a decreasing arithmetic sequence where d is negative. Use tₙ = t₁ + (n − 1)d with n = 18 and d = −150.

Q11 — Find the term number

An arithmetic sequence has t₁ = 8 and d = 11.
Which term number gives a value of 206?

Substitute tₙ = 206, t₁ = 8, and d = 11 into the formula, then solve the equation for n.

Q12 — Tabular reading

A table shows the following arithmetic sequence:
n: 1, 2, 3, 4, 5
tₙ: 12, 17, 22, 27, 32
Using the nth-term rule, what is t₁₀₀?

First identify t₁ and d from the table, then apply the nth-term formula with n = 100.

Q13 — Simple interest loan repayment

A loan of $6 000 is repaid in equal monthly instalments of $450.
The outstanding balance each month forms an arithmetic sequence.
After how many months will the loan be fully repaid (balance = 0)?

Divide the loan amount by the monthly repayment. If the result is not a whole number, round up — you need enough complete payments to clear the full balance.

Q14 — Recursive generation

A sequence is defined recursively: t₁ = 3, tₙ = 2 × tₙ₋₁.
What is t₆?

Start with t₁ = 3 and multiply by 2 repeatedly until you reach the 6th term.

Q15 — Common ratio

The sequence 5, 15, 45, 135, … is geometric.
What is the common ratio r?

The common ratio is found by dividing any term by the term before it.

Q16 — nth-term rule

Using tₙ = t₁ × r^(n−1), find the 8th term of a geometric sequence with t₁ = 2 and r = 3.

Substitute t₁ = 2, r = 3, and n = 8 into the formula tₙ = t₁ × r^(n−1). Calculate 3⁷ first.

Q17 — Bacterial growth

A bacterial culture starts with 500 bacteria and doubles every hour.
How many bacteria are present after 7 hours?

Doubling every hour means r = 2. The population after 7 hours is term 8 in the sequence (since t₁ is the count at hour 0). Use tₙ = t₁ × r^(n−1).

Q18 — Exponential decay / diminishing value

A car is purchased for $24 000 and loses 20% of its value each year (diminishing-value depreciation).
What is its value (to the nearest dollar) after 5 years?

Losing 20% each year means 80% of the value remains, so r = 0.80. Use the geometric nth-term formula with t₁ = $24 000 and n = 6 (end of year 5 is the 6th term).

Q19 — Graphical / exponential curve

A geometric sequence with r > 1 is plotted (term value vs. term number).
Which type of curve best describes the shape of the graph?
Enter: 1 = straight line, 2 = exponential curve, 3 = parabola

Think about what kind of growth a geometric sequence with r > 1 produces — does the value increase by the same amount each step, or does it grow faster and faster?

Q20 — Find the first term

The 4th term of a geometric sequence is 54 and the common ratio is r = 3.
What is the first term t₁?

Substitute t₄ = 54, r = 3, and n = 4 into the formula, then solve for t₁. You'll need to divide by r³.

Q21 — Recursive rule for decay

A radioactive sample decays so that each year only 75% of the previous year's amount remains.
Write the recursive rule as tₙ = r × tₙ₋₁. What is r? (Give your answer rounded to 2 decimal places.)

If 75% of the amount remains each year, what fraction (as a decimal) do you multiply by to get the next term?

Q22 — Compound growth model

An investment of $1 000 grows at 6% per annum compounded annually.
What is the value (in dollars, rounded to 2 decimal places) after 10 years?

Growing at 6% per year means each term is multiplied by 1.06, so r = 1.06. Use tₙ = t₁ × r^(n−1) with t₁ = 1000 and n = 11 (after 10 years of growth).

Q23 — Tabular reading (geometric)

A table shows:
n: 1, 2, 3, 4
tₙ: 4, 12, 36, 108
What is t₇?

First find r by dividing consecutive terms from the table, then identify t₁ and apply the nth-term formula with n = 7.

Q24 — Diminishing-value: which year

Equipment worth $50 000 depreciates at 15% per year (diminishing-value).
What is its value (to the nearest dollar) at the end of year 6?

Depreciating at 15% per year means 85% of the value remains, so r = 0.85. Use the geometric formula with t₁ = $50 000 and n = 7 (end of year 6 is the 7th term).

Q25 — Geometric vs arithmetic identification

Consider the sequence: 2, 6, 18, 54, 162.
The ratio of consecutive terms is constant. What is the value of t₁₀?

Find r by dividing any term by the one before it, then use the geometric nth-term formula with n = 10. You'll need to calculate 3⁹.