Step by step
The question
Question: work out the sum for k = 0 to 4 of (10 − 3k).
Σk = 04(10 − 3k)
1 Read the sigma
Σ is the Greek letter sigma.
It means: add up.
Start with k = 0, and stop at k = 4.
That is 4 − 0 + 1: 5 terms.
For each k we work out 10 − 3k.
Σ = add up
start: k = 0, stop: k = 4
4 − 0 + 1 = 5 terms
Careful!
From 0 to 4 there are 5 terms, not 4: we count both ends.
So 4 − 0 + 1.
Careful: 4 − 0 = 4 is not the number of terms
2 Write the terms
k = 0: we put 0 in place of k.
We get 10.
k = 0: 10 − 3(0) = 10 + 0 = 10
k = 1 gives 7.
k = 1: 10 − 3(1) = 10 − 3 = 7
k = 2: 10 − 3(2) = 10 − 6 = 4
k = 3: 10 − 3(3) = 10 − 9 = 1
And k = 4 gives −2.
That is the last one.
k = 4: 10 − 3(4) = 10 − 12 = −2
3 Add them
Now we add the 5 terms.
Σk = 04(10 − 3k) = 10 + 7 + 4 + 1 − 2
So the sum is 20.
= 20
4 Check
Let's check with the pairs from the sequences lesson.
The terms go up by the same amount, so first plus last, 10 + (−2), times 5 terms, over 2.
Pairs: 5 × (10 + (−2))2 = 5 × 82 = 20
We get 20 again.
It works!
Let's look back.
First: read the sigma.
Then: write the terms.
Then: add them.
Last: check.
Answer
= 20