Step by step
The question
Question: work out the sum for i = 1 to 5 of (4i − 1).
Σi = 15(4i − 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 4i − 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 = 154i − 1 ✗
2 Write the terms
i = 1: we put 1 in place of i.
We get 3.
i = 1: 4(1) − 1 = 4 − 1 = 3
i = 2 gives 7.
i = 2: 4(2) − 1 = 8 − 1 = 7
i = 3: 4(3) − 1 = 12 − 1 = 11
i = 4: 4(4) − 1 = 16 − 1 = 15
And i = 5 gives 19.
That is the last one.
i = 5: 4(5) − 1 = 20 − 1 = 19
3 Add them
Now we add the 5 terms.
Σi = 15(4i − 1) = 3 + 7 + 11 + 15 + 19
So the sum is 55.
= 55
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 + 19, times 5 terms, over 2.
Pairs: 5 × (3 + 19)2 = 5 × 222 = 55
We get 55 again.
It works!
Let's look back.
First: read the sigma.
Then: write the terms.
Then: add them.
Last: check.
Answer
= 55