Methods of Induction and Binomial Theorem — Class 11 Maharashtra HSC Mathematics revision notes
A 4-page visual revision sheet for Methods of Induction and Binomial Theorem, written for Class 11 Maharashtra HSC Mathematics. Formulas, diagrams and the traps that cost marks, laid out to be read the night before rather than the term before. Free, and it opens inside CapCompass.
Page 1 as PDF
Methods of Induction and Binomial Theorem in one look
prove it for one domino, prove it falls onto the next, and every domino after it is proved too
The four steps of induction
- Foundation: prove P(n) is true for n=1 (checking n=2, 3 too is good practice, not required)
- Assumption: ASSUME P(n) is true for n=k. Not proved, just the hypothesis the next step uses
- Succession: using that assumption, PROVE P(n) is true for n=k+1
- Induction: conclude P(n) is true for every positive integer n
Illustration: sum of the first n natural numbers
- the statement P(n)
1+2+3+…+n=n(n+1)/2
Proving a summation formula by induction
Q. prove 1+2+3+…+n=n(n+1)/2 for all nin N
- Foundation, n=1: LHS=1, RHS=1(2)/2=1. Equal, so P(1) is true
- Assumption: suppose 1+2+…+k=k(k+1)/2
- Succession: 1+2+…+k+(k+1)=k(k+1)/2+(k+1)=(k+1)left(k/2+1right)=(k+1)(k+2)/2
- this is exactly P(k+1). By induction, P(n) holds for every nin N
Answer: 1+2+…+n=n(n+1)/2, proved
Page 1 of 4. The full sheet, all 4 pages, opens free in the app.
What this Methods of Induction and Binomial Theorem sheet covers
- The four steps of induction
- Illustration: sum of the first n natural numbers
- the statement P(n)
- Proving a summation formula by induction
- Induction on a more elaborate series
- Induction for divisibility
- Induction on a divisibility statement
- Binomial theorem, positive integer index
- for a,bin R, nin N
- with alternating sign
- Expanding a binomial power
- General and middle terms
- the (r+1)th term of (a+b)^n
- n even: one middle term
- n odd: two middle terms
- Finding a specific term
- Finding the middle term
- Binomial theorem vs binomial series
- positive integer n
- negative or fractional n
- Binomial series, negative or fractional index
- n not a positive integer, |x|<1
- A fractional-index expansion
- Standard expansions, |x|<1
- (1+x) to the -1
- (1-x) to the -1
- (1+x) to the -2
- (1-x) to the -2
- Traps that cost marks
- Tricks that save time
How CapCompass uses this sheet
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