Chapter 2 · Sequences · Concept 3
2-11 Summation Symbol Σ (1)
After a wildfire, conservationists plant 200 saplings, then 300 the next year, then 400 — while drought kills off 40% of what survives each year. Tracking a growing plan against a shrinking one, decade after decade, needs a way to write 'add all of these up' that fits on one line.
Guiding question
How do you write a long sum without writing it all out?
Learning outcome. Use the summation symbol to represent sums, and calculate their values.
Σ just means “add these up”
If the general term of a sequence is aₖ, then the sum from the 1st term to the n-th is written like this:
the letter under Σ
where the counting starts — usually k = 1
the number above Σ
where the counting stops
the bit after Σ
what to work out and add, each time
Σ₁⁴ 7k = 7 + 14 + 21 + 28
Put k = 1, 2, 3, 4 into 7k and add what comes out. That is the whole meaning.
One small habit. When a sequence is written with n in it, change the n to a k before putting it inside Σ — the n is needed for the number on top.
There is nothing hidden inside a Σ. Put the first number in, then the next, and write plus signs between the answers. Watch it happen.
Going the other way needs two things, not one: the general term, and the value of k that produces the last term. People find the first and guess the second.
2 + 5 + 8 + … + 29
2(b) Write 6² + 7² + 8² + … + 15² using Σ.
- 1
Ignore the squares for a moment and look at the bases.
6, 7, 8, …, 15The squaring is the same on every term, so it is not the pattern. The bases are.
Work with a partner. For question 2, one of you finds the general term and the other finds the number on top. Both are needed, and they are different jobs.
1. Write these without Σ.
2. Write these using Σ.
1. Write these without Σ.
2. Write these using Σ.
A startup installs 1 solar panel in week 1, 4 in week 2, 7 in week 3, and (3n − 2) in week n. Write a single expression using Σ for the total number installed from week 1 through week n.
Σ₁³ 2 has no k inside it at all. Does that mean the answer is 2? And what would Σ₁ⁿ 7 come to?
Ask what Σ is actually instructing you to do.
Putting the last VALUE on top of the Σ.
For 4 + 9 + … + 119 the top is 24, not 119. Solve 119 = 5k − 1 to find which k lands on the last term.
Leaving the n inside.
The letter inside Σ is k, and n belongs on top. Writing Σ₁ⁿ (3n − 2) means something different and wrong.
Thinking Σ with no k gives one copy.
Σ₁³ 2 is 2 + 2 + 2 = 6. The k counts the repeats even when it never appears.
Losing brackets on a squared term.
(k + 5)² is not k² + 5. Write the bracket and keep it.
Not testing at k = 1.
Two seconds of checking catches almost every wrong general term before it costs you the question.
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