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The coefficient of x^2 in the expansion of (1+2x)^n is 60. given that n>0, find the value of n.​

The Coefficient Of X2 In The Expansion Of 12xn Is 60 Given That Ngt0 Find The Value Of N class=

Sagot :

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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:

Oladayoademosuidn

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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