Sequences

**Theatre seating.** A theatre has 20 seats in row 1, 23 seats in row 2, 26 seats in row 3, and so on, each row adding 3 seats. (a) How many seats are in row 12? (b) Which row is the first to have at least 50 seats? (c) How many seats are there in total in the first 12 rows?

**Bouncing ball.** A ball is dropped from a height of 80 cm. Each bounce reaches $\frac{3}{4}$ of the previous bounce height. (a) Write the height of the $n$th bounce as a geometric sequence. (b) What is the height of the 5th bounce, to the nearest mm? (c) After how many bounces is the height less than 10 cm for the first time?

**Salary growth.** A graduate has two job offers. - **Company A:** starting salary £24,000, with annual increases of £1,500. - **Company B:** starting salary £22,000, with annual increases of 5% on the previous year's salary. (a) Write a formula for the salary $A_n$ at company A and $B_n$ at company B in year $n$. (b) In which year does B first overtake A? (c) What is the total earned at each company over the first 5 years?

**Shifted pattern.** The diagram below shows the first three patterns made from dots. Pattern 1: 4 dots (square corners) Pattern 2: 7 dots Pattern 3: 10 dots (a) How many dots are in pattern $n$? (b) Pattern $k$ has 88 dots. Find $k$. (c) A student says "Pattern 100 has 304 dots." Is she correct? Show your reasoning.

**Geometric shrinkage.** A piece of paper has area 800 cm². It is folded in half repeatedly so that the new exposed top area halves each time. (a) Write a sequence for the visible area $A_n$ after $n$ folds. (b) After how many folds is the visible area less than 1 cm²? (c) Could the area ever be exactly 0 cm²? Justify mathematically.

**Mixed arithmetic/geometric.** A sequence has $u_1 = 6$. The differences between consecutive terms form a geometric sequence: $u_2 - u_1 = 4$, $u_3 - u_2 = 8$, $u_4 - u_3 = 16$, and so on. (a) Find $u_5$. (b) Find a closed form for $u_n$.

**Card stacking.** Sienna builds a house of cards. Level 1 (top) uses 2 cards. Level 2 needs 5 cards. Level 3 needs 8 cards. Each level adds 3 more cards than the level above. (a) How many cards are needed for level $n$? (b) How many cards are needed to build all levels from 1 to 10? (c) Sienna has 200 cards. Including all levels, what is the largest house she can build?

**Identifying the type.** For each sequence, state whether it is arithmetic, geometric, or neither, and find a formula or rule for $u_n$. (a) $5, 9, 13, 17, 21, \ldots$ (b) $3, 6, 12, 24, 48, \ldots$ (c) $1, 4, 9, 16, 25, \ldots$ (d) $2, 5, 11, 23, 47, \ldots$

**Two sequences meeting.** Sequence A: $u_n = 4n + 3$. Sequence B: starts at 47 and decreases by 2 each term. (a) Write a formula for sequence B. (b) Find the value of $n$ for which $A_n = B_n$. (c) What is the value of the common term?

**Recursive notation challenge.** A sequence is defined by $$u_1 = 1, \quad u_2 = 3, \quad u_{n+1} = u_n + 2 u_{n-1}.$$ (a) Find $u_3$, $u_4$, $u_5$, $u_6$. (b) Show that all terms are odd integers. (c) Calculate $\dfrac{u_{n+1}}{u_n}$ for $n = 1, 2, 3, 4, 5$. What do you notice?

**Sum of squares puzzle.** Consider the sequence $1, 4, 9, 16, 25, \ldots$ of square numbers. (a) Sum the first five terms. (b) The formula $\displaystyle S_n = \frac{n(n+1)(2n+1)}{6}$ gives the sum of the first $n$ square numbers. Verify this for $n = 5$. (c) Find the sum of the first 20 square numbers. (d) Find the sum: $4 + 9 + 16 + 25 + \ldots + 400$.

**Aesthetics of growth.** A nautilus shell grows in a logarithmic spiral. Each chamber is $\phi$ times the previous, where $\phi = \frac{1+\sqrt{5}}{2} \approx 1.618$ (the golden ratio). The smallest chamber has area $1 \text{ mm}^2$. (a) Write a formula for the area $A_n$ of the $n$th chamber. (b) Estimate the area of the 8th chamber (3 s.f.). (c) The total area of the first $n$ chambers is given by a geometric sum. Find the total area of the first 8 chambers (3 s.f.).