MathematicsChapter 7: Mathematical Induction and Binomial Theorem
Linear Sum
P(n): sum = n(n+1)/2.
Learning objectives
- State the rule or meaning of linear sum in easy words.
- Apply linear sum in a clear step-by-step solution.
- Check the conditions, working, and final answer for accuracy.
Definition
P(n): sum = n(n+1)/2. IH: k(k+1)/2. Add k+1: [k(k+1)+2(k+1)]/2 = (k+1)k+22. That is P(k+1).
Formula
sum = n(n+1)/2. IH
Where / symbols
- LINEAR SUM
- P(n): sum = n(n+1)/2. IH: k(k+1)/2. Add k+1: [k(k+1)+2(k+1)]/2 = (k+1)(k+2)/2. That is P(k+1).
- LINEAR SUM
- P(n): sum = n(n+1)/2.
Relationship
- Sk+1 = Sk + ak+1
Overview
Understand the idea, then apply it:
Main Idea
P(n): sum = n(n+1)/2. IH: k(k+1)/2. Add k+1: [k(k+1)+2(k+1)]/2 = (k+1)k+22. That is P(k+1).
Examples Add k+1 · Factor (k+1) · Match formula
Exam Use
Write the rule first: P(n): sum = n(n+1)/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 — LINEAR SUM
P(n): sum = n(n+1)/2. IH: k(k+1)/2. Add k+1: [k(k+1)+2(k+1)]/2 = (k+1)k+22. That is P(k+1).
WHY — Why it matters
This rule is a separate learning unit inside induction on sums. It can be defined, applied, and checked in its own FBISE question.
HOW — How to answer
Write the rule first: P(n): sum = n(n+1)/2. Substitute carefully, show one step per line, and check signs, restrictions, units, and the final result.
How to answer a linear sum 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
Do not assume P(k+1). Derive it from P(k) plus the new term.
Key points
- common denominator
- Sk+1 = Sk + ak+1
- factor after you add
- focused answer
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Linear Sum
Sticky Notes · Quick Revision
Core Idea
- P(n): sum = n(n+1)/2.
What to Know
- P(n): sum = n(n+1)/2. IH: k(k+1)/2. Add k+1: [k(k+1)+2(k+1)]/2 = (k+1)k+22. That is P(k+1).
Example or Evidence
- Add k+1
- Factor (k+1)
- Match formula
How to Answer
- Write the rule first: P(n): sum = n(n+1)/2. Substitute carefully, show one step per line, and check signs, restrictions, units, and the final result.
Common Error
- Do not assume P(k+1). Derive it from P(k) plus the new term.
FBISE Exam Check
- State the rule, show the working, and check the answer.
60-second revision
- P(n): sum = n(n+1)/2.
- Add k+1
- Factor (k+1)
- State the rule, show the working, and check the answer.
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