1. (a) Find the first four terms, in ascending powers of x, of the binomial expansion of (1+8y)^{rac{1}{2}} giving each term in simplest form. (b) Explain how you could use x = rac{1}{32} in the expansion to find an approximation for \sqrt{5}. There is no need to carry out the calculation.

A-Level Edexcel Maths Pure Solving Equations: 1. (a) Find the first four terms

1. (a) Find the first four terms, in ascending powers of x, of the binomial expansion of (1+8y)^{rac{1}{2}} giving each term in simplest form - Edexcel - A-Level Maths Pure - Question 2 - 2020 - Paper 1

Question 2

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1. (a) Find the first four terms, in ascending powers of x, of the binomial expansion of (1+8y)^{rac{1}{2}} giving each term in simplest form. (b) Explain how you ... show full transcript

Worked Solution & Example Answer:1. (a) Find the first four terms, in ascending powers of x, of the binomial expansion of (1+8y)^{rac{1}{2}} giving each term in simplest form - Edexcel - A-Level Maths Pure - Question 2 - 2020 - Paper 1

Step 1

Find the first four terms, in ascending powers of x, of the binomial expansion of (1+8y)^{rac{1}{2}}

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Answer

To find the first four terms of the binomial expansion of ((1 + 8y)^{\frac{1}{2}}), we will use the binomial theorem, which states that:

$(a + b)^n = \sum_{k=0}^{n} {n \choose k} a^{n-k} b^k$

In this case, we have: a = 1, b = 8y, and n = \frac{1}{2}.

Term 1: When k = 0, ${\frac{1}{2} \choose 0} 1^{\frac{1}{2}} (8y)^0 = 1$
Term 2: When k = 1, ${\frac{1}{2} \choose 1} 1^{\frac{1}{2}} (8y)^1 = \frac{1}{2} \cdot 8y = 4y$
Term 3: When k = 2, ${\frac{1}{2} \choose 2} 1^{\frac{1}{2}} (8y)^2 = \frac{\frac{1}{2} \cdot (\frac{1}{2} - 1)}{2!} (8y)^2 = \frac{\frac{1}{2} \cdot (-\frac{1}{2})}{2}(64y^2) = - \frac{16y^2}{2} = -8y^2$
Term 4: When k = 3, ${\frac{1}{2} \choose 3} 1^{\frac{1}{2}} (8y)^3 = \frac{\frac{1}{2} \cdot (\frac{1}{2} - 1)(\frac{1}{2} - 2)}{3!} (512y^3) = \frac{\frac{1}{2} \cdot (-\frac{1}{2})(-\frac{3}{2})}{6}(512y^3) = \frac{96y^3}{6} = 16y^3$

Thus, the first four terms in ascending powers of x are:

$1 + 4y - 8y^2 + 16y^3$

Step 2

Explain how you could use x = \frac{1}{32} in the expansion to find an approximation for \sqrt{5}

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Answer

To approximate \sqrt{5}$, we can substitute (x = \frac{1}{32}) into the expansion derived in part (a).

First, recognize that we can express \sqrt{5}$ in a suitable form:

$\sqrt{5} = \sqrt{4 + 1} = \sqrt{4(1 + \frac{1}{4})} = 2\sqrt{1 + \frac{1}{4}}$

Now we can use the binomial expansion for this case, with: a = 1, b = \frac{1}{4}, and n = \frac{1}{2}:

Once we have the approximation from the expansion, we simply multiply the result by 2 (from the factor outside the square root) to obtain an approximate value for \sqrt{5}.$

1. (a) Find the first four terms, in ascending powers of x, of the binomial expansion of (1+8y)^{ rac{1}{2}} giving each term in simplest form - Edexcel - A-Level Maths Pure - Question 2 - 2020 - Paper 1

Worked Solution & Example Answer:1. (a) Find the first four terms, in ascending powers of x, of the binomial expansion of (1+8y)^{ rac{1}{2}} giving each term in simplest form - Edexcel - A-Level Maths Pure - Question 2 - 2020 - Paper 1

Find the first four terms, in ascending powers of x, of the binomial expansion of (1+8y)^{ rac{1}{2}}

Explain how you could use x = \frac{1}{32} in the expansion to find an approximation for \sqrt{5}

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1. (a) Find the first four terms, in ascending powers of x, of the binomial expansion of (1+8y)^{rac{1}{2}} giving each term in simplest form - Edexcel - A-Level Maths Pure - Question 2 - 2020 - Paper 1

Worked Solution & Example Answer:1. (a) Find the first four terms, in ascending powers of x, of the binomial expansion of (1+8y)^{rac{1}{2}} giving each term in simplest form - Edexcel - A-Level Maths Pure - Question 2 - 2020 - Paper 1

Find the first four terms, in ascending powers of x, of the binomial expansion of (1+8y)^{rac{1}{2}}