Numerical approximation using expansion

1. (a) Find the first 4 terms in ascending powers of \(x\) of the binomial expansion of

$$\left( 2 + \frac { x } { 4 } \right) ^ { 10 }$$ giving each term in its simplest form.
(b) Use your expansion to find an estimated value for \(2.025 ^ { 10 }\), stating the value of \(x\) which you have used and showing your working.

1. (a) Use the binomial theorem to find the first 4 terms, in ascending powers of \(x\), of the expansion of

$$\left( 1 - \frac { x } { 2 } \right) ^ { 8 }$$ Give each term in its simplest form.
(b) Use the answer to part (a) to find an approximate value to \(0.9 ^ { 8 }\) Write your answer in the form \(\frac { a } { b }\) where \(a\) and \(b\) are integers.

1. (a) Find the first 4 terms, in ascending powers of \(x\), of the binomial expansion of

$$\left( 1 + \frac { x } { 3 } \right) ^ { 18 }$$ giving each term in its simplest form.
(b) Use the answer to part (a) to find an estimated value for \(\left( \frac { 31 } { 30 } \right) ^ { 18 }\), stating the value of \(x\) that you have used and showing your working. Give your estimate to 4 decimal places. II

1. (a) Find, in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\), the binomial expansion of

$$\left( 2 + \frac { x } { 8 } \right) ^ { 13 }$$ fully simplifying each coefficient.
(b) Use the answer to part (a) to find an approximation for \(2.0125 ^ { 13 }\) Give your answer to 3 decimal places. Without calculating \(2.0125 { } ^ { 13 }\)
(c) state, with a reason, whether the answer to part (b) is an overestimate or an underestimate.

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. 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.

3. (a) Find the first 4 terms of the binomial expansion, in ascending powers of \(x\), of $$\left( 1 + \frac { x } { 4 } \right) ^ { 8 }$$ giving each term in its simplest form.
(b) Use your expansion to estimate the value of \(( 1.025 ) ^ { 8 }\), giving your answer to 4 decimal places.

8

1. Find the first three terms, in ascending powers of \(x\), of the expansion of \(( 1 - 2 x ) ^ { 10 }\)
   [0pt] [3 marks]
   8
2. Carly has lost her calculator. She uses the first three terms, in ascending powers of \(x\), of the expansion of \(( 1 - 2 x ) ^ { 10 }\) to evaluate \(0.998 ^ { 10 }\)
   Find Carly's value for \(0.998 ^ { 10 }\) and show that it is correct to five decimal places.
   [0pt] [3 marks]

1. (a) Find the first 3 terms, in ascending powers of \(x\), of the binomial expansion of

$$\left( 2 + \frac { 3 x } { 4 } \right) ^ { 6 }$$ giving each term in its simplest form.
(b) Explain how you could use your expansion to estimate the value of \(1.925 ^ { 6 }\) You do not need to perform the calculation.

1. The binomial expansion of

$$( 1 + a x ) ^ { 12 }$$ up to and including the term in \(x ^ { 2 }\) is $$1 - \frac { 15 } { 2 } x + k x ^ { 2 }$$ where \(a\) and \(k\) are constants.

1. Show that \(a = - \frac { 5 } { 8 }\)
2. Hence find the value of \(k\) Using the expansion and making your method clear,
3. find an estimate for the value of \(\left( \frac { 17 } { 16 } \right) ^ { 12 }\), giving your answer to 4 decimal places.

4

1. Expand \(( 1 + x ) ^ { 4 }\).
2. Use your expansion to determine the exact value of \(1002 ^ { 4 }\).

4. (a) Expand \(( 1 + 3 x ) ^ { 8 }\) in ascending powers of \(x\) up to and including the term in \(x ^ { 3 }\). You should simplify each coefficient in your expansion.
(b) Use your series, together with a suitable value of \(x\) which you should state, to estimate the value of (1.003) \({ } ^ { 8 }\), giving your answer to 8 significant figures.

3.

1. Find the first three terms, in ascending powers of \(x\), of the expansion of $$\left( 3 - \frac { x } { 2 } \right) ^ { 8 }$$ [3 marks]
2. Use your expansion to estimate the value of \(2.995 ^ { 8 }\).
   [0pt] [2 marks]

4

1. Find the first three terms in the expansion of \(( 1 - 3 x ) ^ { 4 }\) in ascending powers of \(x\). 4
2. Using your expansion, approximate \(( 0.994 ) ^ { 4 }\) to six decimal places.