Grade 12 Mathematics · Unit 1 · Chapter 1.3 · QuickNotes

The Sigma Notation and Partial Sums

These QuickNotes give you the most important points to remember for the ESSLCE. They are based on the MoE Grade 12 Mathematics Textbook, Unit 1, Chapter 1.3.

~8 min read

Summary

A partial sum S_n is the sum of the first n terms of a sequence.
Sigma notation, written with the symbol ∑, is a short way to write a long sum.
The sum of an arithmetic sequence is S_n = (n ÷ 2)(A₁ + A_n), and the sum of a geometric sequence is S_n = G₁(1 − rⁿ) ÷ (1 − r) when r is not 1.

Key Words

Series: a series is the sum you get when you add the terms of a sequence.
Partial sum (S_n): a partial sum is the sum of the first n terms of a sequence.
Sigma notation: sigma notation uses the Greek letter ∑ to write a sum in a short, neat way.
Index of summation: the index is the counter, often k, that moves up one step at a time.
Lower and upper limits: the lower limit is where the counter starts and the upper limit is where it stops.

What This Chapter Is About

This chapter teaches you how to add the terms of a sequence and how to write that sum in a short way.
You also learn quick formulas for the sum of an arithmetic sequence and the sum of a geometric sequence.

Partial Sums

A partial sum adds the terms of a sequence up to a chosen point, so S₁ = a₁ and S₂ = a₁ + a₂.
In general, S_n = a₁ + a₂ + a₃ + … + a_n is the sum of the first n terms.
The word “partial” reminds you that you have only added part of the sequence, not all of it.

Sigma Notation

Sigma notation uses the symbol ∑ to mean “add up”, so it turns a long sum into a short one.
You write the index counter and its lower limit below the symbol, and the upper limit above it.
The rule on the right tells you what to add, and you put each value of the counter into that rule.

S_n = ∑_k=1ⁿ a_k = a₁ + a₂ + a₃ + … + a_n

This means the counter k starts at the lower limit 1, goes up to the upper limit n, and you add the term a_k at each step.

Worked example: write out ∑_k=1⁴ 3k as a sum.

Put k = 1, 2, 3, 4 into 3k and add: 3 + 6 + 9 + 12 = 30.

Properties of Sigma Notation

A constant factor can be moved out front, so ∑ c·a_k = c·∑ a_k.
A sum or difference can be split into two sums, so ∑ (a_k + b_k) = ∑ a_k + ∑ b_k.
Adding the constant 1 a total of n times gives n, so ∑_k=1ⁿ 1 = n, and ∑_k=1ⁿ c = nc.

Sum of an Arithmetic Sequence

The mathematician Carl Friedrich Gauss found a fast way to add 1 + 2 + 3 + … + 100 while still a young pupil.
He wrote the sum forward, then wrote it backward underneath, and saw that every column adds to the same total.
Because there are n columns and each pair adds to A₁ + A_n, twice the sum is n(A₁ + A_n), so you halve it to get the sum.

Figure 1.3: Gauss wrote 1 + 2 + 3 + 4 + 5 forward and backward, so each column adds to 6. Five columns give 30, and half of that is the sum 15.

The sum of the first n natural numbers is S_n = (n ÷ 2)(n + 1), which is the same idea with A₁ = 1 and A_n = n.

S_n = (n ÷ 2)[2A₁ + (n − 1)d] = (n ÷ 2)(A₁ + A_n)

Use the first form when you know the first term and the common difference, and use the second form when you know the first and last terms.

Worked example: add the first 35 terms of A_n = 5n.

Here A₁ = 5 and A₃₅ = 5(35) = 175. So S₃₅ = (35 ÷ 2)(5 + 175) = (35 ÷ 2)(180) = 35 × 90 = 3150.

Sum of a Geometric Sequence

To add a geometric sequence, you use a formula that depends on the common ratio r.
When r is not 1, the sum of the first n terms is S_n = G₁(1 − rⁿ) ÷ (1 − r).
When r is exactly 1, every term equals G₁, so the sum is simply S_n = nG₁.

S_n = G₁(1 − rⁿ) ÷ (1 − r), for r ≠ 1

This means you subtract rⁿ from 1, multiply by the first term, then divide by (1 − r). If r = 1, use S_n = nG₁ instead.

Note on the textbook: The printed Theorem 1.4 shows the r = 1 case as nG₁ ÷ (r − 1), but that would divide by zero. The correct value when r = 1 is S_n = nG₁, which the unit Summary also gives.

Worked example: add the first 5 terms of 2, 4, 8, 16, 32.

Here G₁ = 2 and r = 2, so S₅ = 2(1 − 2⁵) ÷ (1 − 2) = 2(1 − 32) ÷ (−1) = 2(−31) ÷ (−1) = 62.

Quick Comparison Table

Sum of	Formula to use	When it helps
Arithmetic sequence	S_n = (n ÷ 2)[2A₁ + (n − 1)d]	You know A₁ and d
Arithmetic sequence	S_n = (n ÷ 2)(A₁ + A_n)	You know the first and last terms
Geometric sequence (r ≠ 1)	S_n = G₁(1 − rⁿ) ÷ (1 − r)	The ratio r is not 1
Geometric sequence (r = 1)	S_n = nG₁	Every term is the same

Common Mistakes to Avoid

Do not forget the r = 1 case for a geometric sum, because then you simply use nG₁.
Do not miscount the number of terms n, since one extra or missing term changes the whole sum.
Do not mix up the index limits, because the lower limit is where the counter starts and the upper limit is where it stops.
Do not raise r to the wrong power, because the geometric sum uses rⁿ, not rⁿ⁻¹.

Easy Ways to Remember

For an arithmetic sum, picture the Gauss trick: average the first and last terms, then multiply by how many terms there are.
The big sigma symbol ∑ just shouts “add everything from the bottom number up to the top number”.
For a geometric sum, remember “one minus r to the n, over one minus r”, and switch to nG₁ only when r is 1.

Quiz

Tap an answer to check it.

1. What does ∑_k=1³ 2k equal?

2. What is the partial sum S₃ of the sequence 4, 7, 10, 13, …?

3. Find the sum of the first 50 natural numbers.

4. An arithmetic sequence has A₁ = 4 and d = 5. What is the sum of the first 10 terms?

5. A geometric sequence has G₁ = 3 and r = 2. What is the sum of the first 4 terms?

Remember: A partial sum S_n adds the first n terms, and sigma notation ∑ writes that sum in short. For an arithmetic sequence use S_n = (n ÷ 2)(A₁ + A_n), and for a geometric sequence use S_n = G₁(1 − rⁿ) ÷ (1 − r) when r ≠ 1, or nG₁ when r = 1.

Grade 12 Mathematics Unit 1 : Sequences and Series 1.3 The Sigma Notation and Partial Sums

Grade 12 Mathematics Unit 1 : Sequences and Series