MathematicsChapter 4: Sequences and Series
With Ends
If last term l is given, S_n = n(a+l)/2.
Learning objectives
- State the rule or meaning of with ends in easy words.
- Apply with ends in a clear step-by-step solution.
- Check the conditions, working, and final answer for accuracy.
Definition
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).
Formula
Sn = n(a+l)/2. Find n first from an
Where / symbols
- WITH ENDS
- If last term l is given, S_n = n(a+l)/2. Find n first from a_n = l if needed. Word problems: seats in a theatre, logs in a stack (school).
- WITH ENDS
- If last term l is given, S_n = n(a+l)/2.
Relationship
- Sn = n/2 * (2a+(n-1)d)
Overview
Understand the idea, then apply it:
Main Idea
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).
Examples 1 to n: n(n+1)/2 · Find n from last term · Check a_n
Exam Use
Write the rule first: If last term l is given, S_n = n(a+l)/2. Substitute carefully, show one step per line, and check signs, restrictions, units, and the final result.
Examples State the central idea · Use one accurate detail · Check the final response
WHAT — 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).
WHY — Why it matters
This rule is a separate learning unit inside sum of an arithmetic series. It can be defined, applied, and checked in its own FBISE question.
HOW — How to answer
Write the rule first: If last term l is given, S_n = n(a+l)/2. Substitute carefully, show one step per line, and check signs, restrictions, units, and the final result.
How to answer a with ends question
- Read the exact command word and identify what the question asks.
- State the matching idea, rule, or meaning before adding detail.
- Use the unit evidence or mathematical working that directly supports the answer.
- Check accuracy, relevance, and answer length before finishing.
In simple words
S_n uses n/2, not n. Forgetting /2 doubles the answer.
Key points
- average of ends * n
- Sn = n/2 * (2a+(n-1)d)
- also n/2 * (a1+a_n)
- focused answer
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.With Ends
Sticky Notes · Quick Revision
Core Idea
- If last term l is given, Sn = n(a+l)/2.
What to Know
- 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).
Example or Evidence
- 1 to n: n(n+1)/2
- Find n from last term
- Check a_n
How to Answer
- Write the rule first: If last term l is given, S_n = n(a+l)/2. Substitute carefully, show one step per line, and check signs, restrictions, units, and the final result.
Common Error
- S_n uses n/2, not n. Forgetting /2 doubles the answer.
FBISE Exam Check
- State the rule, show the working, and check the answer.
60-second revision
- If last term l is given, Sn = n(a+l)/2.
- 1 to n: n(n+1)/2
- Find n from last term
- State the rule, show the working, and check the answer.
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