Arithmetic sequences and series AI SL
1. Introduction to Arithmetic Sequences
Definition:
An arithmetic sequence (or arithmetic progression) is a sequence in which the difference between consecutive terms is constant.
This constant difference is the common difference, denoted by 
.
Notation and General Form:
Let:
be the first term,- be the common difference.
Then the sequence is:
where 
is the 
-th term.
Formula for the -th Term:
The -th term is
Example 1:
Given 
, 
. Find 
.
Answer: 
.
2. Sum of an Arithmetic Series
Definition:
An arithmetic series is the sum of terms of an arithmetic sequence.
Sum of the First Terms:
Equivalently, using 
:
![S_n = \dfrac{n}{2}\,\big[\,2u_1 + (n-1)d\,\big]](https://i0.wp.com/alphyschool.org/wp-content/ql-cache/quicklatex.com-d1ae459695c9ed509d8363bd63b433b2_l3.png?resize=215%2C37&ssl=1 "Rendered by QuickLaTeX.com")
Example 2:
Find the sum of the first 20 terms of 
Step 1: 
.
Step 2: 
.
Answer: 
.
3. Sigma Notation for Arithmetic Series
Definition:
Sigma notation uses 
to represent sums. For an arithmetic series:
with 
.
Example 3:
Express the sum of the first 15 terms of 
in sigma notation.
Answer: 
.
4. Applications of Arithmetic Sequences and Series
Arithmetic models appear in finance, physics, CS, and more—whenever change is uniform per step.
Example 4: Salary Increment
Starting salary $40,000; raise $2,500 each year. Total earnings over 10 years?
, 
, 
Step 1 (10th-year salary): 
.
Step 2 (total 10-year earnings): 
.
Answer: $512,500.
5. Analysing, Interpreting, and Predicting Non-Arithmetic Models
Real data may deviate from a perfect arithmetic pattern; checking first differences helps diagnose fit.
Example 5: Population Growth with Irregularity
Population over 5 years: 10,000 → 15,000 → 21,000 → 28,000 → 36,000.
Differences: 
(not constant) → not perfectly arithmetic, but increasing.
Prediction (6th year): If the pattern of +1,000 extra per year continues, next increase ≈ 
:
Answer: 45,000 (model-based estimate).
6. Real-World Examples
Example 6: Loan Repayment
You pay $100 each month for 12 months to clear a $1,000 loan (simple illustration; ignores interest compounding).
- Monthly payment sequence:
(12 terms; 
)
Total paid over 12 months: 
.
Note: The $1,200 already includes the $1,000 principal (plus $200 extra as fees/interest in this simple example). Don’t add $1,000 again.
Example 7: Construction Costs
Costs rise $15,000 each quarter, starting at $100,000. Total over 1 year (4 quarters)?
- Quarterly costs:
Answer: $490,000.
Example 8: Road Construction
Segment lengths increase by 10 km from a 50 km start. Find the 10th segment and total after 10 segments.
10th segment: 
.
Total length: 
.
Example 9: Dividend Payments
Annual dividend starts at $1,200 and increases by $200 each year. Total over 5 years?
![S_5=\dfrac{5}{2}\big(1{,}200 + [1{,}200+(5-1)\cdot 200]\big)=\dfrac{5}{2}(1{,}200+2{,}000)=5\cdot 1{,}600=8{,}000](https://i0.wp.com/alphyschool.org/wp-content/ql-cache/quicklatex.com-f9bf0e85a3c90005f2154b257c2870fe_l3.png?resize=640%2C40&ssl=1 "Rendered by QuickLaTeX.com")
Answer: $8,000.
Example 10: Machine Depreciation
Value starts at $30,000 and decreases by $3,000 per year.
- Initial value
, common difference 
7th term (year-by-year value listing): 
.
Value after 7 full years of depreciation: 
.
Total depreciation over 7 years: 
.
Why two answers? 
is the 7th listed term; “after 7 years” applies 7 decreases. Define your timeline clearly in problems.
Arithmetic sequences and series are powerful models for steady change. Even when data isn’t perfectly arithmetic, checking first differences lets you judge fit and make reasonable predictions.