Binomial Theorem (1 Viewer)

[allGenreGamer]

I have a few questions in the binomial theorem topic which I cannot solve:

- Write down the (k + 2)th term of (a + b)^2n
- Find the coefficient of a^-2 in the expansion of (a+1/a)^3 * (a - 1/a) ^ 5

These questions are from Fitzpatrick - the one who writes the hardest maths problems! I feel that if I can solve these questions, then the coming exam will be no problem, please help me.

 

[CM_Tutor]

First Question:

For the expansion (a + b)²ⁿ, the (k + 1)th term, T_k+1 is T_k+1 = ²ⁿC_ka^2n-kb^k
Replacing (k + 1) with (k + 2), T_k+2 = ²ⁿC_k+1a^2n-(k+1)b^k+1 = ²ⁿC_k+1a^2n-k-1b^k+1

 

[CM_Tutor]

Second Question:

[a + (1 / a)]³ = ³C₀a³ + ³C₁a²(1 / a)¹ + ³C₂a¹(1 / a)² + ³C₃(1 / a)³ = ³C₀a³ + ³C₁a + ³C₂ / a + ³C₃ / a³ ___ (A)

[a - (1 / a)]⁵ = ⁵C₀a⁵ + ⁵C₁a⁴(-1 / a)¹ + ⁵C₂a³(-1 / a)² + ... + ⁵C₅(-1 / a)⁵ = ⁵C₀a⁵ - ⁵C₁a³ + ⁵C₂a - ... - ⁵C₅ / a⁵ ___ (B)

Every term in the expansion of [a + (1 / a)]³[a - (1 / a)]⁵ is formed by multiplying a term in (A) by a term in (B). We seek the term in a^-2. This term is:

Term in a³ in (A) * Term in a^-5 in (B) + Term in a in (A) * Term in a^-3 in (B) + Term in a^-1 in (A) * Term in a^-1 in (B) + Term in a^-3 in (A) * Term in a in (B)
= [³C₀a³ * (-⁵C₅ / a⁵)] + [³C₁a * (⁵C₄ / a³)] + [(³C₂ / a) * (-⁵C₃ / a)] + [(³C₃ / a³) * ⁵C₂a]
= a^-2(-³C₀ * ⁵C₅ + ³C₁ * ⁵C₄ - ³C₂ * ⁵C₃ + ³C₃ * ⁵C₂)
= a^-2(-1 * 1 + 3 * 5 - 3 * 10 + 1 * 10)
= -6a^-2

So, the coefficient of a^-2 in the expansion of [a + (1 / a)]³[a - (1 / a)]⁵ is -6.

 

[allGenreGamer]

Thanks! With that explanation, I've got my head around the problem. Alrite... getting confident for the test!

 

[masteraal]

how retarded does it look on the screen. very hard to explain without writing it on paper eh

 

[rumour]

BT is so hard!!!!!!!
*must study for tomorrow!!!*

 

[Xayma]

Originally posted by masteraal
how retarded does it look on the screen. very hard to explain without writing it on paper eh

Your just lucky he uses html tags. Imagine it without them