Series & Sigma Notation — Challenge
Adding a long list of terms without adding them one at a time.
What these problems 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
What is 1+2+3+⋯+10?
- 45
- 50
- 55
- 100
- 110
Worked answer
55
Pair the first with the last, the second with the second last, and so on: each pair adds to 11.
Ten numbers make five such pairs, giving 5· 11 = 55.
The general rule is n(n+1)2, which gives 10· 112 = 55.
Problem 2
What is the sum 2+4+6+8+10?
- 25
- 30
- 35
- 40
- 50
Worked answer
30
Add them directly, or use the arithmetic series rule.
The average of the first and last is 2+102 = 6, and there are five terms.
5· 6 = 30.
Problem 3
What is ∑k=14k?
- 4
- 6
- 10
- 16
- 24
Worked answer
10
The notation means add the values of k from 1 to 4.
1+2+3+4 = 10.
The product of those numbers would be 24, which is a different operation.
Problem 4
An arithmetic series has 7 terms, the first being 3 and the last 21. What is its sum?
- 72
- 84
- 96
- 120
- 168
Worked answer
84
The sum of an arithmetic series is the number of terms times the average of the first and last.
The average is 3+212 = 12.
7· 12 = 84.
Problem 5
What is the sum of the infinite geometric series 1+12+14+⋯?
- 1
- 1.5
- 2
- 2.5
- It has no finite sum
Worked answer
2
The first term is 1 and each term is half the one before, so the ratio is 12.
When the ratio is between -1 and 1 the series converges to a1-r.
11-12 = 112 = 2. The partial sums climb towards 2 without reaching it.
Problem 6
What is 3+6+9?
- 15
- 18
- 21
- 24
- 27
Worked answer
18
Add in order: 3+6 = 9, then 9+9 = 18.
Alternatively, all three are multiples of 3: 3(1+2+3) = 3· 6 = 18.
Problem 7
What is the sum of the first four odd numbers?
- 10
- 12
- 16
- 20
- 24
Worked answer
16
The first four odd numbers are 1, 3, 5 and 7.
1+3+5+7 = 16.
The sum of the first n odd numbers is always n2, and 42 = 16.
Problem 8
What is the sum of five terms each equal to 7?
- 12
- 30
- 34
- 35
- 57
Worked answer
35
Adding the same number repeatedly is multiplication.
5· 7 = 35.
Problem 9
What is 2+4+8+16+32?
- 30
- 46
- 62
- 64
- 126
Worked answer
62
This is a geometric series with first term 2 and ratio 2.
Adding directly: 2+4 = 6, +8 = 14, +16 = 30, +32 = 62.
Note the sum is always two less than the next term, here 64-2 = 62.