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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 4 - 2008 - Paper 2
Question 4
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 4 - 2008 - Paper 2
Step 1
Find the first 4 terms of the expansion of \( \left( 1 + \frac{x}{2} \right)^{10} \)
Answer
To find the first four terms of the expansion, we use the Binomial Theorem:
[ (1 + a)^n = \sum_{k=0}^{n} \binom{n}{k} a^k ]
In our case, ( a = \frac{x}{2} ) and ( n = 10 ). Thus, we calculate:
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For ( k = 0 ): [ \binom{10}{0} \left(\frac{x}{2}\right)^0 = 1 ]
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For ( k = 1 ): [ \binom{10}{1} \left(\frac{x}{2}\right)^1 = 10 \cdot \frac{x}{2} = 5x ]
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For ( k = 2 ): [ \binom{10}{2} \left(\frac{x}{2}\right)^2 = 45 \cdot \left(\frac{x^2}{4}\right) = \frac{45}{4}x^2 ]
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For ( k = 3 ): [ \binom{10}{3} \left(\frac{x}{2}\right)^3 = 120 \cdot \left(\frac{x^3}{8}\right) = 15x^3 ]
Thus, the first four terms in the expansion are: [1 + 5x + \frac{45}{4}x^2 + 15x^3]
Step 2
Use your expansion to estimate the value of \( (1.005)^{10} \)
Answer
To estimate ( (1.005)^{10} ) using our expansion, we substitute ( x = 10 \cdot 0.005 = 0.05 ) into our previously found expansion:
[1 + 5(0.05) + \frac{45}{4}(0.05)^2 + 15(0.05)^3]
Calculating each term:
- ( 1 = 1 )
- ( 5(0.05) = 0.25 )
- ( \frac{45}{4}(0.0025) = \frac{45}{4} \cdot 0.0025 = 0.028125 )
- ( 15(0.000125) = 0.001875 )
Adding these: [ 1 + 0.25 + 0.028125 + 0.001875 = 1.280000 ]
Therefore, the estimated value of ( (1.005)^{10} ) is approximately ( 1.28000 ).