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Sum of an Arithmetic Series
Sn = n/2 (2a + (n-1)d) = n/2 (first + last). Meaning: n copies of the average of the first and last term. Use the first form if you know a and d; use first+last if you know the ends. Gauss's story: 1+...+100 = 1002 * 101.
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Sum of an Arithmetic Series
What is it?
Sn = n/2 (2a + (n-1)d) = n/2 (first + last). Meaning: n copies of the average of the first and last term. Use the first form if you know a and d; use first+last if you know the ends. Gauss's story: 1+...+100 = 1002 * 101.
In simple words
- Sn = n/2 * (2a + (n-1)d) = n/2 * (first + last).
- Meaning: n copies of the average of the first and last term.
- Use the first form if you know a and d; use first+last if you know the ends.
- S_n uses n/2, not n. Forgetting /2 doubles the answer.
Important terms
SUM OF AN ARITHMETIC SERIES: Sn = n/2 (2a + (n-1)d) = n/2 (first + last). Meaning: n copies of the average of the first and last term. Use the first form if you know a and d; use first+last if you know the ends. Gauss's story: 1+...+100 = 1002 * 101.
1. WITH a AND d: Plug n. Keep /2 till you have an even number if you want integers. Sn is not an.
2. WITH ENDS: If last term l is given, Sn = n(a+l)/2. Find n first from an = l if needed. Word problems: seats in a theatre, logs in a stack (school).
What you should learn
- State the main ideas and notation used in sum of an arithmetic series.
- Apply the rules of sum of an arithmetic series in a clear, step-by-step solution.
- Check signs, conditions, and the final answer for mathematical accuracy.
Examples
- Sum of 10 terms
- d negative: still OK
- S_n grows like n^2
WITH a AND d
Plug n. Keep /2 till you have an even number if you want integers. Sn is not an.
WITH ENDS
If last term l is given, Sn = n(a+l)/2. Find n first from an = l if needed. Word problems: seats in a theatre, logs in a stack (school).
Recommended for Class 11
Full school-level explanation
Sn = n/2 (2a + (n-1)d) = n/2 (first + last). Meaning: n copies of the average of the first and last term. Use the first form if you know a and d; use first+last if you know the ends. Gauss's story: 1+...+100 = 1002 * 101.
Two formulas, same sum:
WHAT — SUM OF AN ARITHMETIC SERIES
Sn = n/2 (2a + (n-1)d) = n/2 (first + last). Meaning: n copies of the average of the first and last term. Use the first form if you know a and d; use first+last if you know the ends. Gauss's story: 1+...+100 = 1002 * 101.
RULE — 1. WITH a AND d
Plug n. Keep /2 till you have an even number if you want integers. Sn is not an.
METHOD — 2. WITH ENDS
If last term l is given, Sn = n(a+l)/2. Find n first from an = l if needed. Word problems: seats in a theatre, logs in a stack (school).
WORKED EXAMPLE
Worked: sum of first 20 terms of 2, 5, 8, ... a=2, d=3, n=20 S20 = 202 * (4 + 19*3) = 10 * (4+57) = 10*61 = 610
WITH a AND d
Plug n. Keep /2 till you have an even number if you want integers. Sn is not an.
Remember
- n/2 times the bracket
Examples
- Sum of 10 terms
- d negative: still OK
- S_n grows like n^2
WITH ENDS
If last term l is given, Sn = n(a+l)/2. Find n first from an = l if needed. Word problems: seats in a theatre, logs in a stack (school).
Remember
- average of ends * n
Examples
- 1 to n: n(n+1)/2
- Find n from last term
- Check a_n
How to answer a sum of an arithmetic series question
How it works
- Identify the given quantities, notation, and required result.
- Choose the matching rule and write it before substitution.
- Work one algebraic step at a time, keeping brackets and signs clear.
- Check restrictions, simplify the result, and verify it where possible.
Remember
- Sn = n/2 * (2a+(n-1)d)
- also n/2 * (a1+a_n)
- 610
- do not drop /2
- n/2 times the bracket
- average of ends * n
- Sn = n/2 * (2a+(n-1)d)
- also n/2 * (a1+a_n)
How the ideas connect
Worked: sum of first 20 terms of 2, 5, 8, ... a=2, d=3, n=20 S20 = 202 * (4 + 19*3) = 10 * (4+57) = 10*61 = 610
Deeper understanding
Go beyond memorising Sum of an Arithmetic Series. Connect the definition to reasons, relationships, exceptions, and exam wording.
How the ideas connect
Worked: sum of first 20 terms of 2, 5, 8, ... a=2, d=3, n=20 S20 = 202 * (4 + 19*3) = 10 * (4+57) = 10*61 = 610
Common mistakes
Exam focus
Core Idea
1. WITH a AND d
2. WITH ENDS
Worked Example
Exam focus
Quick revision
- Sn = n/2 * (2a+(n-1)d)
- also n/2 * (a1+a_n)
- 610
- do not drop /2
- SUM OF AN ARITHMETIC SERIES: Sn = n/2 (2a + (n-1)d) = n/2 (first + last). Meaning: n copies of the average of the first and
- 1. WITH a AND d: Plug n. Keep /2 till you have an even number if you want integers. Sn is not an.
S_n uses n/2, not n. Forgetting /2 doubles the answer.
For long answers on Sum of an Arithmetic Series: define clearly → explain the main parts → give one example → state one exam caution.
Exam Focus
Exam Focus
Important points
- Sn = n/2 * (2a+(n-1)d)
- also n/2 * (a1+a_n)
- 610
- do not drop /2
Important formulas
- Sn = n/2 * (2a+(n-1)d)
- also n/2 * (a1+a_n)
- 610
Common exam traps
- S_n uses n/2, not n. Forgetting /2 doubles the answer.
- Do not use a remembered formula before checking that its conditions fit the question.
Long question structure
Definition → key parts → example → exam caution
Common Mistakes
Common Mistakes
- Mistake
S_n uses n/2, not n. Forgetting /2 doubles the answer.
- Mistake
Do not use a remembered formula before checking that its conditions fit the question.
- Mistake
Do not hide sign changes or restrictions; write them on the line where they are used.
- Mistake
S_n uses n/2, not n. Forgetting /2 doubles the answer.
Revision Zone
60-Second Revision
Quick cues from this topic — use them for last-minute revision.
Sn = n/2 * (2a+(n-1)d)
also n/2 * (a1+a_n)
610
do not drop /2
Sn = n/2 (2a + (n-1)d) = n/2 (first + last). Meaning: n copies of the average of the first and last term. Use the first form if you know a and d; use first
Sn = n/2 * (2a+(n-1)d)
- Definition
- Sn = n/2 (2a + (n-1)d) = n/2 (first + last). Meaning: n copies of the average of the first and last term. Use the first form if you know a and d; use first
- Formula
- Sn = n/2 * (2a+(n-1)d)
- Example
- Sum of 10 terms
- Critical warning
- S_n uses n/2, not n. Forgetting /2 doubles the answer.
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Exam Practice
Define
Q1. Define Sum of an Arithmetic Series.
Explain
Q2. Explain: WITH a AND d.
Example
Q3. Give one clear example of Sum of an Arithmetic Series and say why it fits.
Mistake
Q4. State one common mistake students make about Sum of an Arithmetic Series.
Exam note
Q5. Write a short exam note on Sum of an Arithmetic Series: definition, two points, one example.
Sources & References
School-standard facts for Matric and Intermediate. Original explanations — not copied from textbooks.