Mathematics: Find 3rd term of binomial expansion (4x+y)^5

Mathematics : asked on genyjoannerubiera

29.04.2023

Find 3rd term of binomial expansion (4x+y)^5

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Mathematics

Explain how you can use the terms from the binomial expansion to approximate 4.025. 1st term: 1024 2nd term: 25.6 3rd term: 0.256 4th term: 0.001

Step-by-step answer

P Answered by PhD

add up the given terms: 1024 + 25.6 + 0.256 + 0.001 = 1049.857

Step-by-step explanation:

The 5th power of a binomial is ...

(a+b)^5=a^5+5a^4b+10a^3b^2+10a^2b^3+5ab^4+b^5

For a=4 and b=0.02, the expansion is ...

$4.02^5=4^5+5\cdot 4^4\cdot 0.02+10\cdot 4^3\cdot 0.02^2+10\cdot 4^2\cdot 0.02^3+5\cdot 4\cdot 0.02^4 +0.02^5\\\\=1024+25.6+0.256+0.00128+0.0000032+0.0000000032$

Rounding to thousandths, the series is effectively truncated at 4 terms, so the 5th power of 4.02 is about ...

1024 +25.6 +0.256 +0.001 = 1049.857

Mathematics

Explain how you can use the terms from the binomial expansion to approximate 0.985. 1st term = 1 2nd term = −0.1 3rd term = 0.004 4th term ≈ 0

Step-by-step answer

P Answered by Specialist

There are 5 + 1 = 6 terms in the binomial expansion of (1−0.02)5, and since the 4th term is approximately 0, the 5th and 6th terms are also approximately 0. So, approximate the value of 0.985 by adding the first three terms: 1 + (-0.1) + 0.004 = 0.904.

Mathematics

~explain how you can use the terms from the binomial expansion to approximate 4.02^5 1st term: 1024 2nd term: 25.6 3rd term: 0.256 4th term: 0.001

Step-by-step answer

P Answered by Specialist

See below. The result using the binomial theorem is 1049.8560 compared with 1049.8573 using a calculator.

Step-by-step explanation:

Binomial theorem:-

( a + x)^n = a^n + nC1 a^(n-1) x + nC2 a^(n - 2)x^2 +

(4 + 0.02)^5 = 4^5 + 5C1* 4^4 *0.02 + 5C2* 4^3* 0.02^2 + 5C3*4^2*0.02^3

+ 5C4 * 4 * 0.02^4 + 0.02^5

= 1024 + 5*4^4*0.02 + 10*4^3*0.02^2 + 10*4^2*0.02^3 + 5*4*0.02^4 + 0.02^5

= 1024 + 25.6 + 0.256 + 0.0000032

Mathematics

Explain how you can use the terms from the binomial expansion to approximate 0.985. 1st term = 1 2nd term = −0.1 3rd term = 0.004 4th term ≈ 0

Step-by-step answer

P Answered by Specialist

Mathematics

Explain how you can use the terms from the binomial expansion to approximate 4.025. 1st term: 1024 2nd term: 25.6 3rd term: 0.256 4th term: 0.001

Step-by-step answer

P Answered by PhD

add up the given terms: 1024 + 25.6 + 0.256 + 0.001 = 1049.857

Step-by-step explanation:

The 5th power of a binomial is ...

(a+b)^5=a^5+5a^4b+10a^3b^2+10a^2b^3+5ab^4+b^5

For a=4 and b=0.02, the expansion is ...

$4.02^5=4^5+5\cdot 4^4\cdot 0.02+10\cdot 4^3\cdot 0.02^2+10\cdot 4^2\cdot 0.02^3+5\cdot 4\cdot 0.02^4 +0.02^5\\\\=1024+25.6+0.256+0.00128+0.0000032+0.0000000032$

Rounding to thousandths, the series is effectively truncated at 4 terms, so the 5th power of 4.02 is about ...

1024 +25.6 +0.256 +0.001 = 1049.857

Mathematics

~explain how you can use the terms from the binomial expansion to approximate 4.02^5 1st term: 1024 2nd term: 25.6 3rd term: 0.256 4th term: 0.001

Step-by-step answer

P Answered by Master

See below. The result using the binomial theorem is 1049.8560 compared with 1049.8573 using a calculator.

Step-by-step explanation:

Binomial theorem:-

( a + x)^n = a^n + nC1 a^(n-1) x + nC2 a^(n - 2)x^2 +

(4 + 0.02)^5 = 4^5 + 5C1* 4^4 *0.02 + 5C2* 4^3* 0.02^2 + 5C3*4^2*0.02^3

+ 5C4 * 4 * 0.02^4 + 0.02^5

= 1024 + 5*4^4*0.02 + 10*4^3*0.02^2 + 10*4^2*0.02^3 + 5*4*0.02^4 + 0.02^5

= 1024 + 25.6 + 0.256 + 0.0000032

Mathematics

The coefficient of the second term in the expansion of the binomial (4x+3y) to the 3rd power

Step-by-step answer

P Answered by PhD

You can solve this easily by using Pascal's Triangle (look that up if need be).

Here are the first four rows of P. Triangle:

1
1 1
1 2 1
1 3 3 1

example: expand (a+b)^3.

Look at the 4th row. Borrow and use those coefficients:

1a^3 + 3 a^2b + 3ab^2 + b^3

Now expand (4x+3y)^3:

1(4x)^3 + 3(4x)^2(3y) + 3(4x)*(3y)^2 + (3y)^3

Look at the 2nd term (above):

3(4x)^2(3y) can be re-written as 144x^2y.

The coeff of the 2nd term is 144. Note that (4)^2 = 16

Mathematics

Neil places £2000 in a bank account that pays 1.5% simple interest per year. How much interest will he earn in 6 years?

Step-by-step answer

P Answered by PhD

SI=(P*R*T)/100

P=2000

R=1.5

T=6

SI=(2000*1.5*6)/100

=(2000*9)/100

=180

Neil will earn interest of 180