Induction On Sums

Induction On Sums

What is it?

Classic: 1+2+...+n = n(n+1)/2. 1^2+...+n^2 = n(n+1)2n+16. Geometric 1+r+...+r^n-1 if the book asks. Pattern: Sk+1 = Sk + next term. Replace S_k by the formula from IH, then factor. Meaning: the algebra is 'add one more piece and it still matches the formula with n=k+1'.

In simple words

• Classic: 1+2+...+n = n(n+1)/2.
• 1²+...+n² = n(n+1)2n+16.
• Geometric 1+r+...+r^n-1 if the book asks.
• Do not assume P(k+1). Derive it from P(k) plus the new term.

Important terms

INDUCTION ON SUMS: Classic: 1+2+...+n = n(n+1)/2. 1^2+...+n^2 = n(n+1)2n+16. Geometric 1+r+...+r^n-1 if the book asks. Pattern: Sk+1 = Sk + next term. Replace S_k by the formula from IH, then factor. Meaning: the algebra is 'add one more piece and it still matches the formula with n=k+1'.

1. 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).

2. OTHER SUMS: Cubes, odds, 2ⁿ-1, whatever the paper gives. Always isolate the extra term. Keep one extra = one line.

What you should learn

• State the main ideas and notation used in induction on sums.
• Apply the rules of induction on sums in a clear, step-by-step solution.
• Check signs, conditions, and the final answer for mathematical accuracy.

Examples

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

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).

OTHER SUMS

Cubes, odds, 2ⁿ-1, whatever the paper gives. Always isolate the extra term. Keep one extra = one line.

Full school-level explanation

Classic: 1+2+...+n = n(n+1)/2. 1^2+...+n^2 = n(n+1)2n+16. Geometric 1+r+...+r^n-1 if the book asks. Pattern: Sk+1 = Sk + next term. Replace S_k by the formula from IH, then factor. Meaning: the algebra is 'add one more piece and it still matches the formula with n=k+1'.

Add the next term:

WHAT — INDUCTION ON SUMS

Classic: 1+2+...+n = n(n+1)/2. 1^2+...+n^2 = n(n+1)2n+16. Geometric 1+r+...+r^n-1 if the book asks. Pattern: Sk+1 = Sk + next term. Replace S_k by the formula from IH, then factor. Meaning: the algebra is 'add one more piece and it still matches the formula with n=k+1'.

RULE — 1. 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).

METHOD — 2. OTHER SUMS

Cubes, odds, 2ⁿ-1, whatever the paper gives. Always isolate the extra term. Keep one extra = one line.

WORKED EXAMPLE

Worked: 1+2+...+n = n(n+1)/2, the step. left for k+1 = k(k+1)/2 + (k+1) = (k+1) [k/2 + 1] = (k+1)k+22 right side of P(k+1)

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).

Remember

• common denominator

Examples

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

OTHER SUMS

Cubes, odds, 2ⁿ-1, whatever the paper gives. Always isolate the extra term. Keep one extra = one line.

Remember

• one extra term

Examples

• Write S_k + extra
• Use IH on S_k
• Tidy

How to answer a induction on sums 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

• S_k+1 = S_k + a_k+1
• factor after you add
• factor k+1
• derive, do not assume
• common denominator
• one extra term
• S_k+1 = S_k + a_k+1
• factor after you add

How the ideas connect

Worked: 1+2+...+n = n(n+1)/2, the step. left for k+1 = k(k+1)/2 + (k+1) = (k+1) [k/2 + 1] = (k+1)k+22 right side of P(k+1)

Deeper understanding

Go beyond memorising Induction On Sums. Connect the definition to reasons, relationships, exceptions, and exam wording.

How the ideas connect

Worked: 1+2+...+n = n(n+1)/2, the step. left for k+1 = k(k+1)/2 + (k+1) = (k+1) [k/2 + 1] = (k+1)k+22 right side of P(k+1)

Common mistakes

Exam focus

Core Idea

1. LINEAR SUM

2. OTHER SUMS

Worked Example

Exam focus

Quick revision

• S_k+1 = S_k + a_k+1
• factor after you add
• factor k+1
• derive, do not assume
• INDUCTION ON SUMS: Classic: 1+2+...+n = n(n+1)/2. 1^2+...+n^2 = n(n+1)2n+16. Geometric 1+r+...+r^n-1 if the book a
• 1. 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).

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

For long answers on Induction On Sums: define clearly → explain the main parts → give one example → state one exam caution.