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 the 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
s = a/(1-r)
30 = a/(1-(2/3))
30( 1/3 ) = a
a = 10
Simplify and write with a rational denominator.
4/\sqrt2
4/\sqrt2 = 4/\sqrt2 * \sqrt2/\sqrt2
4/\sqrt2 = (4\sqrt2)/2
4/\sqrt2 = 2\sqrt2
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
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
Sn = n/2(a+tn)
S₁₂ = 12/2(7 + 29)
Sn = 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.
\sqrt(x + 31) = x + 1
x + 31 = (x + 1)²
x + 31 = x2 + 2x + 1
0 = x² + x − 30
0 = (x + 6)(x − 5)
Candidates: x = −6 and x = 5.
Verify:
x = −6: \sqrt((-6)+31) = 5, but x + 1 = −5. Reject.
x = 5: \sqrt(5+31) = 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 formula.
19 should be 18.
The common difference is 4.
tₙ = 6 + 4(n − 1)
tₙ = 4n + 2
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
A mini theatre has 6 rows. The first row has 8 seats, and each row after it has 2 more seats than the last row.
One teammate says there are 48 seats. Another says 78. Who is right? Show the sum.
78 seats is correct.
tₙ = a + (n-1)d
t₆ = 8 + (6 − 1)(2)
t₆ = 18
Sₙ = n/2(a+tₙ)
S₆ = 6/2(8 + 18)
S₆ = 78
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
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 between 20 and 50.
How many possible codes are there?
Hint: Start by finding the first multiple of 6 after 20.
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?
$144 is correct.
First-year common ratio = 0.90
Second-year ratio = 0.80
t₁ = 200(0.90) = 180
t₂ = 180(0.8)2-1 = 144
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 …
s = a/(1-r)
32 = a/(1-(1/2))
a = 16
16 + 8 + 4 + 2 + …
The series converges because |r| < 1.
SAME OR DIFFERENT?
A: 6/\sqrt3
B: 2\sqrt3
Two teammates say these are different answers. Are they right? Rationalize A to settle it.
They are the same answer.
6/\sqrt3 = (6/\sqrt3)(\sqrt3/\sqrt3)
6/\sqrt3 = (6\sqrt3)/3
6/\sqrt3 = 2\sqrt3