3. (a) Find the first 4 terms of the expansion of $ \left( 1 + \frac{x}{2} \right)^{10} $ in ascending powers of $ x $, giving each term in its simplest form. (b) Use your expansion to estimate the value of $(1.005)^{10}$, giving your answer to 5 decimal places.

A-Level Edexcel Maths Pure Simultaneous Equations: 3. (a) Find the first 4 terms of

3. (a) Find the first 4 terms of the expansion of $ \left( 1 + \frac{x}{2} \right)^{10} $ in ascending powers of $ x $, giving each term in its simplest form - Edexcel - A-Level Maths Pure - Question 5 - 2008 - Paper 2

Question 5

$3.-(a)-Find-the-first-4-terms-of-the-expansion-of-$-\left(-1-+-\frac{x}{2}-\right)^{10}-$-in-ascending-powers-of-$-x-$,-giving-each-term-in-its-simplest-form-Edexcel-A-Level Maths Pure-Question 5-2008-Paper 2.png$

3. (a) Find the first 4 terms of the expansion of $ \left( 1 + \frac{x}{2} \right)^{10} $ in ascending powers of $ x $, giving each term in its simplest form. (... show full transcript

Worked Solution & Example Answer:3. (a) Find the first 4 terms of the expansion of $ \left( 1 + \frac{x}{2} \right)^{10} $ in ascending powers of $ x $, giving each term in its simplest form - Edexcel - A-Level Maths Pure - Question 5 - 2008 - Paper 2

Step 1

Find the first 4 terms of the expansion of $ \left( 1 + \frac{x}{2} \right)^{10} $

96%

114 rated

Answer

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 our case, ( a = 1 ), ( b = \frac{x}{2} ), and ( n = 10 ).

Term 1 (k=0): ${10 \choose 0} (1)^{10} \left( \frac{x}{2} \right)^{0} = 1$
Term 2 (k=1): ${10 \choose 1} (1)^{9} \left( \frac{x}{2} \right)^{1} = 10 \cdot \frac{x}{2} = 5x$
Term 3 (k=2): ${10 \choose 2} (1)^{8} \left( \frac{x}{2} \right)^{2} = 45 \cdot \left( \frac{x^2}{4} \right) = \frac{45}{4}x^2 \text{ or } 11.25x^2$
Term 4 (k=3): ${10 \choose 3} (1)^{7} \left( \frac{x}{2} \right)^{3} = 120 \cdot \left( \frac{x^3}{8} \right) = 15x^3$

Thus, the first four terms of the expansion are:

$1 + 5x + \frac{45}{4}x^2 + 15x^3$

Step 2

Use your expansion to estimate the value of $(1.005)^{10}$

99%

104 rated

Answer

To estimate ((1.005)^{10}), we can substitute ( x = 0.01 ) into our binomial expansion as follows:

$\left( 1 + \frac{0.01}{2} \right)^{10} \approx 1 + 5(0.01) + \frac{45}{4}(0.01)^2 + 15(0.01)^3$

First term: ( 1 )
Second term: ( 5(0.01) = 0.05 )
Third term: ( \frac{45}{4}(0.01)^2 = \frac{45}{4} \times 0.0001 = 0.001125 )
Fourth term: ( 15(0.01)^3 = 15 \times 0.000001 = 0.000015 )

Thus, the estimated value is: $1 + 0.05 + 0.001125 + 0.000015 \approx 1.051140$

Rounding to five decimal places, we find: ( (1.005)^{10} \approx 1.05114 )

3. (a) Find the first 4 terms of the expansion of \( \left( 1 + \frac{x}{2} \right)^{10} \) in ascending powers of \( x \), giving each term in its simplest form - Edexcel - A-Level Maths Pure - Question 5 - 2008 - Paper 2

Worked Solution & Example Answer:3. (a) Find the first 4 terms of the expansion of \( \left( 1 + \frac{x}{2} \right)^{10} \) in ascending powers of \( x \), giving each term in its simplest form - Edexcel - A-Level Maths Pure - Question 5 - 2008 - Paper 2

Find the first 4 terms of the expansion of \( \left( 1 + \frac{x}{2} \right)^{10} \)

Use your expansion to estimate the value of \((1.005)^{10}\)

Join the A-Level students using SimpleStudy...