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Problem-solving Pack

Ecolint Campus des NationsMathematics
Year 10 · Sequences

Problem-solving Pack

Name: _________________________________
Date: _________________ Class: ___________

These problems are designed to challenge you. Read each question carefully. Show all your reasoning — a correct answer without working receives no credit.

1Problem 1 of 12
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?

Working space

2Problem 2 of 12
Bouncing ball. A ball is dropped from a height of 80 cm. Each bounce reaches 34\frac{3}{4} of the previous bounce height.

(a) Write the height of the nnth 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?

Working space

3Problem 3 of 12
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 AnA_n at company A and BnB_n at company B in year nn.
(b) In which year does B first overtake A?
(c) What is the total earned at each company over the first 5 years?

Working space

4Problem 4 of 12
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 nn?
(b) Pattern kk has 88 dots. Find kk.
(c) A student says "Pattern 100 has 304 dots." Is she correct? Show your reasoning.

Working space

5Problem 5 of 12
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 AnA_n after nn 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.

Working space

6Problem 6 of 12
Mixed arithmetic/geometric. A sequence has u1=6u_1 = 6. The differences between consecutive terms form a geometric sequence: u2−u1=4u_2 - u_1 = 4, u3−u2=8u_3 - u_2 = 8, u4−u3=16u_4 - u_3 = 16, and so on.

(a) Find u5u_5.
(b) Find a closed form for unu_n.

Working space

7Problem 7 of 12
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 nn?
(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?

Working space

8Problem 8 of 12
Identifying the type. For each sequence, state whether it is arithmetic, geometric, or neither, and find a formula or rule for unu_n.

(a) 5,9,13,17,21,…5, 9, 13, 17, 21, \ldots
(b) 3,6,12,24,48,…3, 6, 12, 24, 48, \ldots
(c) 1,4,9,16,25,…1, 4, 9, 16, 25, \ldots
(d) 2,5,11,23,47,…2, 5, 11, 23, 47, \ldots

Working space

9Problem 9 of 12
Two sequences meeting. Sequence A: un=4n+3u_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 nn for which An=BnA_n = B_n.
(c) What is the value of the common term?

Working space

10Problem 10 of 12
Recursive notation challenge. A sequence is defined by

u1=1,u2=3,un+1=un+2un−1.u_1 = 1, \quad u_2 = 3, \quad u_{n+1} = u_n + 2 u_{n-1}.


(a) Find u3u_3, u4u_4, u5u_5, u6u_6.
(b) Show that all terms are odd integers.
(c) Calculate un+1un\dfrac{u_{n+1}}{u_n} for n=1,2,3,4,5n = 1, 2, 3, 4, 5. What do you notice?

Working space

11Problem 11 of 12
Sum of squares puzzle. Consider the sequence 1,4,9,16,25,…1, 4, 9, 16, 25, \ldots of square numbers.

(a) Sum the first five terms.
(b) The formula Sn=n(n+1)(2n+1)6\displaystyle S_n = \frac{n(n+1)(2n+1)}{6} gives the sum of the first nn square numbers. Verify this for n=5n = 5.
(c) Find the sum of the first 20 square numbers.
(d) Find the sum: 4+9+16+25+…+4004 + 9 + 16 + 25 + \ldots + 400.

Working space

12Problem 12 of 12
Aesthetics of growth. A nautilus shell grows in a logarithmic spiral. Each chamber is ϕ\phi times the previous, where ϕ=1+52≈1.618\phi = \frac{1+\sqrt{5}}{2} \approx 1.618 (the golden ratio).

The smallest chamber has area 1 mm21 \text{ mm}^2.

(a) Write a formula for the area AnA_n of the nnth chamber.
(b) Estimate the area of the 8th chamber (3 s.f.).
(c) The total area of the first nn chambers is given by a geometric sum. Find the total area of the first 8 chambers (3 s.f.).

Working space