Series & Sigma Notation — Practice
Adding a long list of terms without adding them one at a time.
What these questions cover
These questions use 5 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.
- Ratio — A comparison of two quantities, like 3 cups flour to 2 cups water.
- Common difference — The constant added each time, in an arithmetic sequence.
- Common ratio — The constant multiplied each time, in a geometric sequence.
- Series — The sum of the terms of a sequence.
- Limit — The value a function approaches as the input approaches some number.
Problem 1
Evaluate ∑k=14 k.
Worked answer
10
1+2+3+4=10
Problem 2
Evaluate ∑k=13 2k.
Worked answer
12
2+4+6=12
Problem 3
Evaluate ∑k=14 (k+3).
Worked answer
22
4+5+6+7=22
Problem 4
How many terms are there from 5 to 20 counting by 5?
Worked answer
4
5, 10, 15, 20
Four terms
Problem 5
Is 3, 9, 27, 81 arithmetic or geometric? Answer with one word.
Worked answer
geometric
Each is triple the last
Problem 6
Find the sum of the whole numbers from 1 to 50.
Worked answer
1275
502(1+50)
=25 × 51 = 1275
Problem 7
Find the sum of the first 20 terms of 3, 7, 11, …
Worked answer
820
a20=3+19(4)=79
202(3+79)=10 × 82=820
Problem 8
How many terms are there from 7 to 31 counting by 3?
Worked answer
9
31-73=8 gaps
8+1=9 terms
Problem 9
Find the sum of the first 5 terms of 2, 6, 18, …
Worked answer
242
21-2431-3=2 × 121=242
Problem 10
Find the sum of the first 4 terms of 5, 10, 20, …
Worked answer
75
5+10+20+40=75
Problem 11
Find the common ratio of 48, 12, 3, …
Worked answer
1/4
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Problem 12
Find the sum of 8+4+2+1+⋯
Worked answer
16
r=12
81-12=16
Problem 13
Find the sum of 9+3+1+⋯
Worked answer
13.5
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Problem 14
Which infinite series has a finite sum?
- 1+12+14+⋯
- 1+2+4+⋯
- 1+1+1+⋯
- 2+4+8+⋯
Worked answer
1+12+14+⋯
Only the first has |r|<1
The others grow without limit
Problem 15
Write 0.4 as a fraction in lowest terms.
Worked answer
4/9
0.41-0.1=0.40.9=49
Problem 16
The first term is 6 and the sum of the first 10 terms is 195. Find the common difference.
Worked answer
3
195=5(12+9d)
39=12+9d
9d=27 so d=3