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Geometric Sequences and Series · Paper 1 — Free Maths AI SL Practice Questions

1FoundationSAQ-SDefinition and general term of geometric sequences5 marksPaper 1~8 min
The first three terms of a geometric sequence are 66, 1818, and 5454.
(a)
State the common ratio, rr, of the sequence. [1 mark]
(b)
Find the value of the 8th term, u8u_8. [2 marks]
(c)
Find the smallest value of nn such that the sum of the first nn terms exceeds 59 00059\,000. [2 marks]
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Solutions

2FoundationSAQ-SDefinition and general term of geometric sequences7 marksPaper 1~11 min
A population of bacteria grows according to a geometric sequence. On day 1, there are 200200 bacteria. On day 3, there are 800800 bacteria.
(a)
Find the common ratio, rr, of the sequence. [2 marks]
(b)
Find the total number of bacteria counted over the first 6 days. [2 marks]
(c)
A laboratory container can support a maximum of 500 000500\,000 bacteria. Determine the first day on which the number of bacteria on that day exceeds the container's capacity. [3 marks]
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Solutions

3MasterySAQ-SDefinition and general term of geometric sequences5 marksPaper 1~8 min
A new savings account is opened with an initial deposit of USD 2000. At the end of each year, interest is added so that the amount in the account at the start of each year forms a geometric sequence. The amount in the account at the start of the third year is USD 2247.20.
(a)
Show that the common ratio of the geometric sequence is 1.061.06. [2 marks]
(b)
Hence find the amount in the account at the start of the sixth year. Give your answer correct to the nearest dollar. [2 marks]
(c)
The account holder states: "By the start of the tenth year, the total interest earned will have exceeded USD 1000." Determine whether this statement is correct. [1 mark]
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Solutions

4ChallengeSAQ-LDefinition and general term of geometric sequences8 marksPaper 1~12 min

Data

un=u1rn−1u_n = u_1 r^{n-1} Sn=u1(rn−1)r−1,r≠1S_n = \frac{u_1(r^n - 1)}{r - 1}, \quad r \neq 1 S∞=u11−r,∣r∣<1S_\infty = \frac{u_1}{1 - r}, \quad |r| < 1
A fractal pattern is created by drawing squares. The largest square has side length 64 cm64\,\text{cm}. Each subsequent square is drawn inside the previous one, with its vertices at the midpoints of the sides of the previous square, so the side length of each new square is 12\dfrac{1}{\sqrt{2}} times the side length of the previous square.
(a)
Determine the side length of the 66th square. The areas of the squares form a geometric sequence with first term u1=4096 cm2u_1 = 4096\,\text{cm}^2 and common ratio r=12r = \dfrac{1}{2}. [3 marks]
(b)
Determine the total area of the first 1010 squares. The pattern continues indefinitely. The artist claims that the total area of all the squares is less than 9000 cm29000\,\text{cm}^2. [3 marks]
(c)
Determine whether the artist's claim is correct. [2 marks]
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Solutions

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