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Sagot :
n=6
Answer:
Solution given:
The coefficient of x^2=60
we have,
[tex](a+x)^n=C(n,0)a^n+C(n,1)a^(n-1)x+C(n,2)a^(n-2)x^2+C(n,3)a^(n-3)x^3....C(n,r)a^(n-r)x^r+C(n,n)x^n[/tex]
we get x² in 3rd term,so
3rd term of (1+2x)^n is C(n,2)*1^n-2)(2x)^2=C(n,2)4x²
since 1^n is 1.
we have a coefficient of x^2 is 60, so
C(n,2)4=60
C(n,2)=60/4
[tex]\frac{n!}{(n-2)!*2!}[/tex]=15
[tex]\frac{n(n-1)(n-2)!}{(n-2)!*2!}[/tex]=15
n(n-1)=15*2
n^2-n=30
n^2-n-30=0
doing middle term factorization
n^2-6x+5x-30=0
n(n-6)+5(n-6)=0
(n-6)(n+5)=0
either n-6=0
n=6
or,
n+5=0
n=-5 since n>0
so neglected.
So the value of n is 6.
Step-by-step explanation:
The value of n is 6.
Binomial expansion
Since the expression is (1 + 2x)ⁿ, it is a binomial expression.
The general term is ⁿCₓ(1)ⁿ⁻ˣ(2x)ˣ = [ⁿCₓ(2)ˣ]xˣ
Now, for the term in x², xˣ = x². ⇒ x = 2
So, the coefficient of x² is [ⁿC₂(2)²] = 4ⁿC₂
Coefficient of x²
Since the coefficient of x² is 60, we have that
4[ⁿC₂] = 60
Dividing through by 4, we have
ⁿC₂ = 15
n(n - 1)/2! = 15
n(n - 1) = 15 × 2
n(n - 1) = 30
Simplifying
The value of n
n² - n - 30 = 0
n² + 5n - 6n - 30 = 0
Factorizing, we have
n(n + 5) - 6(n + 5) = 0
(n + 5)(n - 6) = 0
n + 5 = 0 or n - 6 = 0
n = -5 or n = 6
since n > 0, n = 6.
So, the value of n is 6.
Learn more about binomial expansion here:
https://brainly.com/question/13602562
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