Use the binomial theorem to write the binomial expansion of

Use the binomial theorem to write the binomial expansion of (1/2x + 3y)^4. Use the binomial theorem to find the 18th term in the binomial expansion of (2x - ysquareroot2)^27. Find the 69th number in the 72nd row (n = 72) of Pascal\'s triangle. In Pascal\'s triangle,_nC_r +_nC_r+1 =_n+1C_r+1. Let X, Y, and Z be three consecutive number in the n^th row of Pascal\'s Triangle, such that X =_nC_r, Y =_nC_r+1, and_nC_r+2. Use the nature of the Pascal\'s Triangle to show that X + Y + Z =_n+2C_r+2 -_nC_r+1.

Solution

10 ) (x/2 +3y)^4

Use the binnomial expansion :(a +b)^n = nC0a^n +nC1a^(n-1)b^1 +nC2 a^(n-2) ......

we also know nCr = n!/r!(n-r)!

(x/2 +3y)^4 = 4C0(x/2)^4(3y)^0 + 4C1(x/2)^3(3y)^2 +4C2(x/2)^2(3y)^2 + 4C3(x/2)^1(3y)^3 +4C4(3y)^4

= x^4/16 + 4(x^3/8)(3y) + 6(x^2/4)(9y^2) + 4(x/2)(27y^3) + 81y^4

= x^4/16 + 3x^3y/2 + 27x^2y^2/2 + 54xy^3 +81y^4

11) (2x - ysqrt2 )^27

18th term :

use the binomail expansion formula: for k th term :(a+b)^n = nCk(a)^n-k(b)^k

Plug in the formula : n = 27 ; k =18 a = 2x ; b = -ysqrt2

18th term : 27C18(2x)^7(-ysqrt2)^18 = 4686825(128x^7)(y^182^9)

= 307155763200x^7y^18