Find the middle term of the expansion of (2x - ⅓x)⁸
Answers
Answer:
1120x⁸/81
Step-by-step explanation:
To Find :-
Middle term in the expansion of (2x - 1/3x)⁸.
How To Do :-
As the power of the expansion is '8' , then the total terms in the expansion is '9'. As '9' is the odd number , So there will be only one middle term in expansion. So by using the formula of middle term in expansion if 'n' is even we can find , which term is the middle term. After obtaining that by using the general term of binomial expansion formula we can find the value of that term.
Formula Required :-
1) If the expansion is '(x - a)ⁿ' then the number of terms in the expansion is 'n + 1'.
2) Middle term in the binomial expansion (x - a)ⁿ if 'n' is even then :-
the middle term is '[(n/2) + 1]'.
3) General term of Binomial expansion (x - a)ⁿ :-
tᵣ ₊ ₁ = (-1)ʳ ⁿCᵣ xⁿ ⁻ ʳ aʳ
4) ⁿCᵣ = n!/(n-r)! (r)!
Solution :-
Total number of terms in the expansion of (2x - 1/3x)⁸ is :-
'n' = 8
= 8 + 1
= 9
Middle term in the expansion is :-
[(n/2)+1]th term :- [ ∴ n = 8]
→ [(8/2) + 1]th term
= [4 + 1]th term
= 5th term
5th term in the expansion of '(2x - 1/3x)⁸ :-
→ tᵣ ₊ ₁ = t₅
→ r + 1 = 5
r = 5 - 1
r = 4
→ t₅ = (-1)⁴ ⁸C₄ (2x)⁸⁻⁴ (x/3)⁴
= 1 × ⁸C₄ × (2x)⁴ × x⁴/3⁴
= ⁸C₄ × 16x⁴ × x⁴/81
= ⁸C₄ × 16x⁸/81
= 8!/(8-4)!(4)! × 16x⁸/81
= 8 × 7 × 6 × 5 × 4!/((4)! × 4!) × 16x⁸/81
= 8 × 7 × 6 × 5/4 × 3 × 2 × 1 × 16x⁸/81
= 7 × 2 × 5 × 16 × x⁸/81
= 1120x⁸/81
∴ Middle term in the expansion of (2x - ⅓x)⁸ = 1120x⁸/81.