Words
Which is a sequence and which is a series? Explain the difference.
A: 7, 10, 13, …
B: 7 + 10 + 13 + …
A is a sequence: an ordered list of terms.
B is a series: a sum of terms.
State the first term and common difference.
23, 19, 15, 11, …
a = 23
d = 19 − 23
d = −4
An item loses 9% of its value each year. What multiplier gives the next year’s value?
r = 1 − 0.09
r = 0.91
The item keeps 91% of its value each year.
Find the first term of this infinite geometric series.
S∞ = 30, r = 2/3
a = S∞(1 − r)
a = 30(1 − 2/3)
a = 30(1/3)
a = 10
Simplify and write with a rational denominator.
4/√2
4/√2 = (4/√2)(√2/√2)
4/√2 = 4√2/2
4/√2 = 2√2
What makes a sequence arithmetic or geometric? Identify each pattern.
A: 20, 15, 10, …
B: 2, 6, 18, …
A is arithmetic: the difference stays the same.
d = 15 − 20
d = −5
B is geometric: the ratio stays the same.
r = 6/2
r = 3
Write a general term for this sequence.
−9, −7, −5, −3, …
tₙ = a + (n − 1)d
tₙ = −9 + 2(n − 1)
tₙ = 2n − 11
Write a general term for this sequence.
5, −10, 20, −40, …
a = 5
r = −10/5
r = −2
tₙ = arⁿ⁻¹
tₙ = 5(−2)ⁿ⁻¹
Find the sum of the first four terms of this geometric sequence.
a = 2, r = 3
S₄ = 2(3⁴ − 1)/(3 − 1)
S₄ = 2(80)/2
S₄ = 80
Check: 2 + 6 + 18 + 54 = 80
Solve. State both solutions.
(x − 6)(2x + 3) = 0
x − 6 = 0
x = 6
2x + 3 = 0
2x = −3
x = −3/2
Both solutions are valid.
Write the first four terms of this infinite geometric series. Does it converge or diverge? Explain.
a = 16, r = −1/2
16 − 8 + 4 − 2 + …
It converges because |r| = 1/2 < 1.
Find the sum of the first 12 terms.
tₙ = 2n + 5
a = 2(1) + 5
a = 7
t₁₂ = 2(12) + 5
t₁₂ = 29
S₁₂ = (12/2)(7 + 29)
S₁₂ = 216
Find the first term of the geometric sequence.
t₄ = −216, r = −3
tₙ = arⁿ⁻¹
−216 = a(−3)³
−216 = −27a
a = 8
Find the first term of this finite geometric series.
Sₙ = −30, r = 0.5, tₙ = −2
Sₙ = (rtₙ − a)/(r − 1)
−30 = [0.5(−2) − a]/(0.5 − 1)
15 = −1 − a
a = −16
Solve and check each possible solution.
√(x + 31) = x + 1
x + 31 = (x + 1)²
0 = x² + x − 30
0 = (x + 6)(x − 5)
Candidates: x = −6 and x = 5.
x = −6: √25 = 5, but x + 1 = −5. Reject.
x = 5: √36 = 6 and x + 1 = 6. Valid.
Only x = 5 is a solution.
ODD ONE OUT
A: 3, 6, 9, 12, …
B: 4, 8, 16, 32, …
C: 10, 7, 4, 1, …
Which pattern is different? Explain using common differences or ratios.
B is geometric: r = 2.
A and C are arithmetic.
A has d = 3; C has d = −3.
FIX THE PATTERN
6, 10, 14, 19, 22, …
One term is wrong. Repair it, then write a general term for the corrected sequence.
19 should be 18.
The common difference is 4.
tₙ = 6 + 4(n − 1)
tₙ = 4n + 2
THE SIGN SWITCH
6, −12, 24, −48, …
A student says the common ratio is 2. Agree or disagree? Give the correct ratio and the next term.
Disagree: the ratio is −2.
r = −12/6
r = −2
Next term = (−48)(−2)
Next term = 96
FRONT-ROW SEATS
A mini theatre has 6 rows. The first row has 8 seats, and each row after it has 2 more seats.
One teammate says there are 48 seats. Another says 78. Who is right? Show the sum.
78 seats is correct.
t₆ = 8 + (6 − 1)(2)
t₆ = 18
S₆ = (6/2)(8 + 18)
S₆ = 78
48 counts 8 seats in every row and misses the extra seats.
SPOT THE LOST SOLUTION
A student solves x(x − 5) = 0 by dividing both sides by x. Their only answer is x = 5.
Which solution did they lose? Give both solutions.
They lost x = 0.
x = 0 OR x − 5 = 0
x = 0 OR x = 5
Both make the original product zero. Dividing by x would exclude x = 0.
TWO TRUTHS AND A LIE
A: A sequence is an ordered list of terms.
B: An arithmetic sequence has a constant difference.
C: Every infinite geometric series with a negative ratio diverges.
Find the lie and fix it.
C is the lie.
An infinite geometric series converges when −1 < r < 1.
For example, r = −1/2 gives a convergent series.
A negative ratio alone does not tell you whether it converges.
UNLOCK THE DOOR
The key code is a multiple of 6 strictly between 20 and 50.
How many possible codes are there? Show how the first and last possible codes help you count.
There are 5 possible codes.
First = 24; last = 48; d = 6
48 = 24 + (n − 1)6
24 = 6(n − 1)
4 = n − 1
n = 5
Check: 24, 30, 36, 42, 48.
PRICE DROP
A $200 game console loses 10% of its value in the first year, then 20% in the second year.
Which value is correct after two years: $140, $144, or $162? Show the two multipliers.
$144 is correct.
First-year multiplier = 0.90
Second-year multiplier = 0.80
V = 200(0.90)(0.80)
V = 144
The second loss is based on the value after the first year.
BUILD A TOTAL OF 32
Your infinite geometric series must have ratio r = 1/2.
What first term makes its sum exactly 32? Write the first four terms as a sum, followed by …
The first term is 16.
a = S∞(1 − r)
a = 32(1 − 1/2)
a = 16
16 + 8 + 4 + 2 + …
The series converges because |r| < 1.
SAME OR DIFFERENT?
A: 6/√3
B: 2√3
Two teammates say these are different answers. Are they right? Rationalize A to settle it.
They are the same answer.
6/√3 = (6/√3)(√3/√3)
6/√3 = 6√3/3
6/√3 = 2√3