Worked examples
The question
Question: work out the sum for i = 1 to 5 of 2i.
Σi = 152i
1 Read the sigma
Σ is the Greek letter sigma.
It means: add up.
Start with i = 1, and stop at i = 5.
That is 5 − 1 + 1: 5 terms.
For each i we work out 2i.
Σ = add up
start: i = 1, stop: i = 5
5 − 1 + 1 = 5 terms
2 Write the terms
i = 1: we put 1 in place of i.
We get 2.
i = 1: 2(1) = 2
i = 2 gives 4.
i = 2: 2(2) = 4
i = 3: 2(3) = 6
i = 4: 2(4) = 8
And i = 5 gives 10.
That is the last one.
i = 5: 2(5) = 10
3 Add them
Now we add the 5 terms.
Σi = 152i = 2 + 4 + 6 + 8 + 10
So the sum is 30.
= 30
4 Check
Let's check with the pairs from the sequences lesson.
The terms go up by the same amount, so first plus last, 2 + 10, times 5 terms, over 2.
Pairs: 5 × (2 + 10)2 = 5 × 122 = 30
We get 30 again.
It works!
Let's look back.
First: read the sigma.
Then: write the terms.
Then: add them.
Last: check.
▶ Watch it solved
The question
Question: work out the sum for i = 1 to 5 of (2i + 1).
Σi = 15(2i + 1)
1 Read the sigma
Σ is the Greek letter sigma.
It means: add up.
Start with i = 1, and stop at i = 5.
That is 5 − 1 + 1: 5 terms.
For each i we work out 2i + 1.
Σ = add up
start: i = 1, stop: i = 5
5 − 1 + 1 = 5 terms
Careful!
The + 1 is inside the bracket.
So it is added in every term, 5 times, not just once.
Careful: not Σi = 152i + 1 ✗
2 Write the terms
i = 1: we put 1 in place of i.
We get 3.
i = 1: 2(1) + 1 = 2 + 1 = 3
i = 2 gives 5.
i = 2: 2(2) + 1 = 4 + 1 = 5
i = 3: 2(3) + 1 = 6 + 1 = 7
i = 4: 2(4) + 1 = 8 + 1 = 9
And i = 5 gives 11.
That is the last one.
i = 5: 2(5) + 1 = 10 + 1 = 11
3 Add them
Now we add the 5 terms.
Σi = 15(2i + 1) = 3 + 5 + 7 + 9 + 11
So the sum is 35.
= 35
4 Check
Let's check with the pairs from the sequences lesson.
The terms go up by the same amount, so first plus last, 3 + 11, times 5 terms, over 2.
Pairs: 5 × (3 + 11)2 = 5 × 142 = 35
We get 35 again.
It works!
Let's look back.
First: read the sigma.
Then: write the terms.
Then: add them.
Last: check.
▶ Watch it solved
The question
Question: write 2 + 4 + 8 + 16 + 32 using sigma notation.
2 + 4 + 8 + 16 + 32
1 Find the rule
The rule from the sequences lesson, with i for the term number: the first term, times 2 to the power (i − 1).
r = 2: ai = 2 × 2i − 1
2 times 2 to the power (i − 1) is 2 to the power i: one more 2.
ai = 2 × 2i − 1 = 2i
2 Count the terms
i counts the terms: 1, 2, 3, 4, 5.
There are 5 terms.
So i starts at 1 and stops at 5.
i = 1, 2, 3, 4, 5
Careful!
The stop number on Σ is the last i, 5.
It is not the last term, 32.
Careful: not Σi = 1322i ✗
3 Write it with Σ
So we write: the sum for i = 1 to 5 of 2 to the power i.
= Σi = 152i
4 Check
Let's check: put i = 1, 2, up to 5 into 2 to the power i.
We get each term of the list.
i = 1 to 5: 2 + 4 + 8 + 16 + 32
That is the sum we started with.
It works!
Let's look back.
First: find the rule.
Then: count the terms.
Then: write it with Σ.
Last: check.
▶ Watch it solved
