Binomial Theorem and Sequences & Series
Binomial Theorem and Sequences & Series tests expansion of (a+b)^n, general and middle terms, and arithmetic and geometric progressions with sums and means.
Core rules
- (a+b)^n=sum from r=0 to n of nCr a^(n-r)b^r. The general term is T(r+1)=nCr a^(n-r)b^r.
- The middle term is T((n+2)/2) when n is even; two middle terms when n is odd.
- For an AP, nth term a+(n-1)d and sum n/2[2a+(n-1)d].
- For a GP, nth term ar^(n-1), sum a(1-r^n)/(1-r) for r≠1, and infinite sum a/(1-r) when |r|<1. AM≥GM for positive numbers.
Worked example
Find the coefficient of x^3 in (2+x)^6. The general term is 6Cr 2^(6-r)x^r. For x^3 set r=3. The coefficient is 6C3 times 2^3=20×8=160. Do not confuse the binomial coefficient 20 with the full coefficient 160.
Traps
- The term number is r+1, not r.
- In (a-b)^n, include the sign (-1)^r.
- The GP sum formula changes at r=1; the infinite sum needs |r|<1.
- AM=GM only when all positive numbers are equal.
- For an AP, the common difference can be negative; keep signs carefully.