Linear Sum

Linear Sum

What is it?

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).

In simple words

• P(n): sum = n(n+1)/2.
• Add k+1
• Factor (k+1)
• State the rule, show the working, and check the answer.

Important terms

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).

LINEAR SUM: P(n): sum = n(n+1)/2.

What you should learn

• 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.

Examples

• Add k+1
• Factor (k+1)
• Match formula

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).

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.

Full school-level explanation

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).

Understand the idea, then apply it:

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.

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).

Remember

• common denominator

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.

Remember

• focused answer
• accurate support

Examples

• State the central idea
• Use one accurate detail
• Check the final response

How to answer a linear sum question

How it works

• 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.

Remember

• common denominator
• S_k+1 = S_k + a_k+1
• factor after you add
• derive, do not assume
• common denominator
• focused answer
• accurate support
• common denominator

Deeper understanding

Go beyond memorising Linear Sum. Connect the definition to reasons, relationships, exceptions, and exam wording.

Common mistakes

Exam focus

Core Idea

What to Know

Example or Evidence

How to Answer

Exam focus

Quick revision

• common denominator
• S_k+1 = S_k + a_k+1
• factor after you add
• derive, do not assume
• 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).
• LINEAR SUM: P(n): sum = n(n+1)/2.

Do not assume P(k+1). Derive it from P(k) plus the new term.

For long answers on Linear Sum: define clearly → explain the main parts → give one example → state one exam caution.