12 multiple-choice questions, progressively harder.
An arithmetic sequence has first term 121212 and common difference −4-4−4. What is the first term that is negative?
Solution
Correct answer: B
Write the term as an=12−4(n−1)=16−4na_n = 12 - 4(n-1) = 16 - 4nan=12−4(n−1)=16−4n and find where it drops below zero.
a4=16−16=0,a5=16−20=−4a_4 = 16 - 16 = 0, \quad a_5 = 16 - 20 = -4a4=16−16=0,a5=16−20=−4
The fourth term is still 000, so the first negative term is a5=−4a_5 = -4a5=−4.
For an=3n−2a_n = 3n - 2an=3n−2, which term equals 100100100?
Correct answer: C
Set the formula equal to 100100100 and solve for nnn.
3n−2=100⇒3n=102⇒n=343n - 2 = 100 \Rightarrow 3n = 102 \Rightarrow n = 343n−2=100⇒3n=102⇒n=34
Two arithmetic sequences both start at a1=5a_1 = 5a1=5. The first has d=3d = 3d=3 and the second has d=4d = 4d=4. For which term number nnn do their nnnth terms differ by 101010?
Correct answer: A
Their nnnth terms are 5+3(n−1)5 + 3(n-1)5+3(n−1) and 5+4(n−1)5 + 4(n-1)5+4(n−1), so the gap between them is (4−3)(n−1)=n−1(4 - 3)(n-1) = n - 1(4−3)(n−1)=n−1.
n−1=10⇒n=11n - 1 = 10 \Rightarrow n = 11n−1=10⇒n=11
How many terms are in the sequence 8,13,18,…,1088, 13, 18, \ldots, 1088,13,18,…,108?
Here a1=8a_1 = 8a1=8 and d=5d = 5d=5. Set the last term equal to 108108108 and solve for nnn.
8+5(n−1)=108⇒5(n−1)=100⇒n=218 + 5(n-1) = 108 \Rightarrow 5(n-1) = 100 \Rightarrow n = 218+5(n−1)=108⇒5(n−1)=100⇒n=21
In a stack of logs, the bottom row has 252525 logs and each row up has 111 fewer. If the top row has 111 log, how many rows are there?
Correct answer: D
The row counts are arithmetic with a1=25a_1 = 25a1=25 and d=−1d = -1d=−1. Set the row count equal to 111.
25−(n−1)=1⇒n−1=24⇒n=2525 - (n-1) = 1 \Rightarrow n - 1 = 24 \Rightarrow n = 2525−(n−1)=1⇒n−1=24⇒n=25
An arithmetic sequence has a4=22a_4 = 22a4=22 and d=−3d = -3d=−3. Find the first term a1a_1a1.
Back up three steps from the fourth term, a1=a4−3da_1 = a_4 - 3da1=a4−3d, keeping the sign of ddd.
a1=22−3×(−3)=22+9=31a_1 = 22 - 3\times(-3) = 22 + 9 = 31a1=22−3×(−3)=22+9=31
The sequence 7,11,15,19,…7, 11, 15, 19, \ldots7,11,15,19,… can be written an=4n+ca_n = 4n + can=4n+c for a constant ccc. Find ccc.
Expand the explicit formula: an=a1+(n−1)d=7+4(n−1)=4n+3a_n = a_1 + (n-1)d = 7 + 4(n-1) = 4n + 3an=a1+(n−1)d=7+4(n−1)=4n+3.
c=3c = 3c=3
You can check it: a1=4(1)+3=7a_1 = 4(1) + 3 = 7a1=4(1)+3=7, matching the first term.
The 3rd term of an arithmetic sequence is 121212 and the 11th term is 444444. What is the common difference?
There are 11−3=811 - 3 = 811−3=8 steps between the terms.
d=44−1211−3=328=4d = \frac{44 - 12}{11 - 3} = \frac{32}{8} = 4d=11−344−12=832=4
An arithmetic sequence has an=50−6na_n = 50 - 6nan=50−6n. What is the last term that is at least 101010?
Require the term to be at least 101010 and find the largest allowed nnn.
50−6n≥10⇒n≤6.6750 - 6n \ge 10 \Rightarrow n \le 6.6750−6n≥10⇒n≤6.67
So n=6n = 6n=6 is the last position that qualifies, and a6=50−36=14a_6 = 50 - 36 = 14a6=50−36=14.
A sequence is defined by a1=100a_1 = 100a1=100 and the recursive rule an=an−1−8a_n = a_{n-1} - 8an=an−1−8. What is a10a_{10}a10?
The recursive rule subtracts 888 each step, so this is arithmetic with d=−8d = -8d=−8. Use 999 steps.
a10=100+9×(−8)=100−72=28a_{10} = 100 + 9\times(-8) = 100 - 72 = 28a10=100+9×(−8)=100−72=28
An arithmetic sequence has a3=10a_3 = 10a3=10 and a8=30a_8 = 30a8=30. Find a20a_{20}a20.
First find ddd over the 8−3=58 - 3 = 58−3=5 steps between the known terms.
d=30−108−3=4d = \frac{30 - 10}{8 - 3} = 4d=8−330−10=4
Then step from a8a_8a8 up 121212 more places to a20=a8+12da_{20} = a_8 + 12da20=a8+12d.
a20=30+12×4=78a_{20} = 30 + 12\times 4 = 78a20=30+12×4=78
An arithmetic sequence has a1=2a_1 = 2a1=2 and d=5d = 5d=5. Is 200200200 a term of the sequence?
Set 2+5(n−1)=2002 + 5(n-1) = 2002+5(n−1)=200 and solve for nnn.
5(n−1)=198⇒n−1=39.65(n-1) = 198 \Rightarrow n - 1 = 39.65(n−1)=198⇒n−1=39.6
Since nnn is not a whole number, 200200200 is not a term.
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