Methods of Induction and Binomial Theorem

A four

A four-step proof method: Step 1 (Foundation) - prove P(1) is true; Step 2 (Assumption) - assume P(k) is true; Step 3 (Succession) - prove P(k+1) is t

(a+b)^n = ∑(r=0 to n) nCr

(a+b)^n = ∑(r=0 to n) nCr × a^(n-r) × b^r = nC0×a^n + nC1×a^(n-1)×b + nC2×a^(n-2)×b^2 + ... + nCn×b^n. The expansion has (n+1) terms with coefficients

The (r+1)th term in (a+b)^n

The (r+1)th term in (a+b)^n is tr+1 = nCr × a^(n-r) × b^r, where 0 ≤ r ≤ n. This formula allows finding any specific term without expanding the entire

If n is even

If n is even: one middle term is the (n/2 + 1)th term. If n is odd: two middle terms are the ((n+1)/2)th and ((n+3)/2)th terms. Example: In (x+y)^6, m

For |x| < 1 and any

For |x| < 1 and any real number n: (1+x)^n = 1 + nx + n(n-1)x^2/2! + n(n-1)(n-2)x^3/3! + ... This gives infinite series. Example: (1+x)^(-1) = 1 - x +