NYJC H2 Mathematics Formula Summary Cheatsheet (very complete, old syllabus)

H2 Mathematics (9740) Chapter 1: Binomial Theorem H2 Mathematics Summarised Formulae Page 1 of 19 Last updated 19 October Chapter 1: Binomial Theorem Expansion of linear algebraic factors 1 2 2( ) ... ... 12 n n n n n r r n n n na b a a b a b a b b r where n is a positive integer and ! !( )!    n n r r n r Expansion of quadratic functions 22 1 2 2 2 2 ( ) [ ( )] ( ) ( ) ...12 nn n n n ax bx c c bx ax nnc c x x c x x Express the quadratic factor into the form (a + b)n where a is the constant and b would be x and x2. Expansion of (1 + x)n 2( 1) ( 1)...( 1)(1 ) 1 ... ... 1 2! ! nr n n n n n rx nx x x x r To expand (1 + kx), replace x with kx. Expansion of (a + bx)n Ascending powers of x (or descending 1/x) Descending powers of x         2 1 11 2! 1 1 ... 1... ... !                                  n n n n n r bax a bb ba bx a x a nnxxan aa xn n n r r b a Range of validity: 1b a a ax x xa b b b      (small x values) (closer to 0 gives better estimate)           2 1 11. 1 ..2! 1 ... 1 ...!                                 n n n n r nn aa bx bx bx nn abx bx aa bx bxbx n n n n x r r a b Range of validity: 1 or a a a a x x xbx b b b      (large x values) Chapter 2: Sequences and Series Simple definitions 11uS [1st term = sum of 1st term] 1;2n n nu S S n    [The difference in 2 sums gives the term] Arithmetic progression (A.P.) Geometric progression (G.P.) Definitions difference of any two successive members of the sequence is a constant each term after the first is found by multiplying the previous one by a fixed non-zero number called the common ratio. General term ( 1)nu a n d   where a = first term, n = no. of terms, d = common difference. 1n nu ar  where r = common ratio 1n n nT S S  Proving 1nnd u u  a non-zero constant 1 n n ur u   is a constant independent of n. Sum of first n terms  212 n nS a n d    2 n nS a l l = last term  1 1 n n ar S r     1 1 n n ar S r    1 aS r    where 0,1r exists 1Sr  Sigma Notation 12 ... b r aa a r a bu u u u u        where a and b are integers. n = (b – a + 1) i.e. upper limit – lower limit + 1. Rules of sigma notation 1 n r a an     1 G.P. 1r r a r     11 nn r r r r au a u     1 1 1 n n n rr r r r r ru v u v           1 11 n n m r m r r r r ru u v        Method of Differences Note: Watch out for method of differences when there is a small difference in the terms being subtracted. Also note that the cancelling is symmetrical, i.e. cancelling the 2nd term on the LHS will cancel the last 2nd term on the RHS. Limit of sequence 11, 0 !nn  0na  (if 0 < a < 1) 1 0na  (if a > 1) The other terms, e.g. an, an where a > 1 will tend to infinit