What's a series
what is the difference between sequence and series
(What is) A sequence is a list of numbers. A series adds them together.
Is 1 + 2 + 3 + 4 + ... geometric?
Answer: (what is) no, the numbers don't multiply by the same ammount each time
What does it mean if a series converges?
(what is) it adds up to a real number, the sum approaches a limit
All geometric series with a ratio less than 1 will converge.
(what is) True, because the terms get smaller and approach zero
If you save $5 the first week and double your saving each week, how much do you save in week 4?
(what is) $40, because 5 × 2 × 2 × 2 = 40 in week 4
What does this series look like? 2 + 4 + 6 + 8 + ...
(What is) It’s an arithmetic series (adds 2 each time).
Is 1 + 2 + 4 + 8 + 16 geometric?
(what is) yes, each term multiplied by 2
What does it mean if a series diverges?
(what is) it keeps growing or doesn’t settle, no fixed sum
The series 2 + 4 + 8 + 16 + 32 is arithmetic because each term after the first doubles the previous term.
(what is) False, because doubling means it’s geometric, not arithmetic.
A bacteria colony triples every hour. If you start with 2 bacteria, how many will there be after 5 hours?
(what is) 486 bacteria, because 2 × 3⁵ = 486 total
What does this series look like? 3 + 6 + 12 + 24 + ...
(What is) A geometric series (multiplies by 2 each time).
What is the common ratio in 81 + 27 + 9 + 3?
(what is) 1/3, each term is one third of the last one
Does the series 5 + 2.5 + 1.25 + ... converge or diverge?
(what is) converge, because the terms get smaller and add up
A geometric series always has to start with a positive number.
(what is) False, because the first term can be any real number, positive or negative.
You start with 100 candies and eat one-third every day. How many candies are left after 2 days?
(what is) about 44 candies, because 100 × (2/3) × (2/3) = 44
What’s the 5th term in this series? 5 + 10 + 20 + ...
(What is) 80
Is the series 5 + 15 + 45 + 135 geometric, and what is its common ratio?
(what is) yes, the common ratio is 3, each term triples
Does the series 1 + 2 + 4 + 8 + ... converge or diverge?
(what is) diverge, because the terms grow without limit
A geometric series with ratio 1 repeats the same term forever.
(what is) True, because multiplying by 1 keeps terms constant
A ball bounces back to half its height every time. If it starts at 16 meters, how high on the 3rd bounce?
(what is) 2 meters, because 16 × (1/2)³ = 2
What do you call a series that never ends?
(What is)An infinite series
Is the series 500 + 150 + 45 + 13.5 geometric? If yes, what is the common ratio?
(what is) yes, the common ratio is 0.3, each term multiplied by 0.3 or 3/10
Does the series 500 + 500 + 500 + ... converge or diverge?
(what is) diverge, because terms don’t get smaller
If a geometric series has ratio -1, its terms switch between positive and negative forever.
(what is) True, because multiplying by -1 flips the sign every term
You add 3 marbles to a jar the first day, then add 2 more marbles than the day before each day. How many marbles added on day 5?
Answer: (what is) 11 marbles, because 3 + 2×4 = 11 on day 5