Step by step
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.
Answer
= 35