1.04a Binomial expansion: (a+b)^n for positive integer n

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

$$\left( 2 - \frac { x } { 8 } \right) ^ { 10 }$$ giving each term in its simplest form. $$\mathrm { f } ( x ) = \left( 2 - \frac { x } { 8 } \right) ^ { 10 } ( a + b x ) , \text { where } a \text { and } b \text { are constants }$$ Given that the first two terms, in ascending powers of \(x\) in the series expansion of \(\mathrm { f } ( x )\), are 256 and \(352 x\),
(b) find the value of \(a\),
(c) find the value of \(b\).

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

6. (a) Find, in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\), the binomial expansion of $$\left( 1 + \frac { 1 } { 4 } x \right) ^ { 12 }$$ giving each term in its simplest form.
(b) Hence find the coefficient of \(x\) in the expansion of $$\left( 3 + \frac { 2 } { x } \right) ^ { 2 } \left( 1 + \frac { 1 } { 4 } x \right) ^ { 12 }$$

5. (a) Find the first 4 terms, in ascending powers of \(x\), of the binomial expansion of $$\left( 3 - \frac { a x } { 2 } \right) ^ { 5 }$$ where \(a\) is a positive constant. Give each term in its simplest form. Given that, in the expansion, the coefficient of \(x\) is equal to the coefficient of \(x ^ { 3 }\),
(b) find the exact value of \(a\) in its simplest form.

11. \(\mathrm { f } ( x ) = ( a - x ) ( 3 + a x ) ^ { 5 }\), where \(a\) is a positive constant

1. Find the first 3 terms, in ascending powers of \(x\), in the binomial expansion of $$( 3 + a x ) ^ { 5 }$$ Give each term in its simplest form. Given that in the expansion of \(\mathrm { f } ( x )\) the coefficient of \(x\) is zero,
2. find the exact value of \(a\).

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

$$\left( 1 - \frac { 1 } { 2 } x \right) ^ { 10 }$$ giving each term in its simplest form.
(b) Hence find the coefficient of \(x ^ { 3 }\) in the expansion of $$\left( 3 + 5 x - 2 x ^ { 2 } \right) \left( 1 - \frac { 1 } { 2 } x \right) ^ { 10 }$$

2. One of the terms in the binomial expansion of \(( 3 + a x ) ^ { 6 }\), where \(a\) is a constant, is \(540 x ^ { 4 }\)

1. Find the possible values of \(a\).
2. Hence find the term independent of \(x\) in the expansion of $$\left( \frac { 1 } { 81 } + \frac { 1 } { x ^ { 6 } } \right) ( 3 + a x ) ^ { 6 }$$

4. (a) Find the first three terms, in ascending powers of \(x\), of the binomial expansion of $$( 2 + p x ) ^ { 6 }$$ where \(p\) is a constant. Give each term in simplest form. Given that in the expansion of $$\left( 3 - \frac { 1 } { 2 } x \right) ( 2 + p x ) ^ { 6 }$$ the coefficient of \(x ^ { 2 }\) is \(- \frac { 3 } { 4 }\) (b) find the possible values of \(p\). \includegraphics[max width=\textwidth, alt={}, center]{52c90d0e-a5e4-45fa-95a4-9523287e7588-11_2255_50_314_34}

3. (a) Find the first 4 terms, in ascending powers of \(x\), of the binomial expansion of $$\left( 2 - \frac { k x } { 4 } \right) ^ { 8 }$$ where \(k\) is a non-zero constant. Give each term in simplest form. $$f ( x ) = ( 5 - 3 x ) \left( 2 - \frac { k x } { 4 } \right) ^ { 8 }$$ In the expansion of \(\mathrm { f } ( x )\), the constant term is 3 times the coefficient of \(x\).
(b) Find the value of \(k\).
T

1. Find the coefficient of the term in \(x ^ { 7 }\) of the binomial expansion of

$$\left( \frac { 3 } { 8 } + 4 x \right) ^ { 12 }$$ giving your answer in simplest form.

4. (a) Find the first four terms, in ascending powers of \(x\), of the binomial expansion of $$\left( 2 - \frac { 1 } { 4 } x \right) ^ { 6 }$$ (b) Given that \(x\) is small, so terms in \(x ^ { 4 }\) and higher powers of \(x\) may be ignored, show $$\left( 2 - \frac { 1 } { 4 } x \right) ^ { 6 } + \left( 2 + \frac { 1 } { 4 } x \right) ^ { 6 } = a + b x ^ { 2 }$$ where \(a\) and \(b\) are constants to be found.

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.

1. The binomial expansion, in ascending powers of \(x\), of

$$( 3 + p x ) ^ { 5 }$$ where \(p\) is a constant, can be written in the form $$A + B x + C x ^ { 2 } + D x ^ { 3 } \ldots$$ where \(A\), \(B\), \(C\) and \(D\) are constants.

1. Find the value of \(A\) Given that
   • \(B = 18 D\)
   • \(p < 0\)
   • find
     1. the value of \(p\)
     2. the value of \(C\)

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

$$\left( 1 - \frac { 1 } { 6 } x \right) ^ { 9 }$$ giving each term in simplest form.
(b) Hence find the coefficient of \(x ^ { 3 }\) in the expansion of $$( 10 x + 3 ) \left( 1 - \frac { 1 } { 6 } x \right) ^ { 9 }$$ giving the answer in simplest form.

3. (a) Find the first 4 terms, in ascending powers of \(x\), in the binomial expansion of $$\left( 1 + \frac { x } { 4 } \right) ^ { 12 }$$ giving each coefficient in its simplest form.
(b) Find the term independent of \(x\) in the expansion of $$\left( \frac { x ^ { 2 } + 8 } { x ^ { 5 } } \right) \left( 1 + \frac { x } { 4 } \right) ^ { 12 }$$

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) Hence find the constant term in the series expansion of $$\left( 3 - \frac { 1 } { x } \right) ^ { 2 } \left( 2 - \frac { x } { 4 } \right) ^ { 10 }$$

1. The first three terms, in ascending powers of \(x\), of the binomial expansion of \(( 1 + k x ) ^ { 16 }\) are

$$1 , - 4 x \text { and } p x ^ { 2 }$$ where \(k\) and \(p\) are constants.

1. Find, in simplest form,
   1. the value of \(k\)
   2. the value of \(p\) $$g ( x ) = \left( 2 + \frac { 16 } { x } \right) ( 1 + k x ) ^ { 16 }$$ Using the value of \(k\) found in part (a),
2. find the term in \(x ^ { 2 }\) in the expansion of \(\mathrm { g } ( x )\). $$\begin{aligned} u _ { 1 } & = 6 \\ u _ { n + 1 } & = k u _ { n } + 3 \end{aligned}$$ where \(k\) is a positive constant.
3. Find, in terms of \(k\), an expression for \(u _ { 3 }\) Given that \(\sum _ { n = 1 } ^ { 3 } u _ { n } = 117\)
4. find the value of \(k\).

1. A curve \(C\) has equation \(y = \mathrm { f } ( x )\) where

$$f ( x ) = ( 2 - k x ) ^ { 5 }$$ and \(k\) is a constant.
Given that when \(\mathrm { f } ( x )\) is divided by \(( 4 x - 5 )\) the remainder is \(\frac { 243 } { 32 }\)

1. show that \(k = \frac { 2 } { 5 }\)
2. Find the first three terms, in ascending powers of \(x\), of the binomial expansion of $$\left( 2 - \frac { 2 } { 5 } x \right) ^ { 5 }$$ giving each term in simplest form. Using the solution to part (b) and making your method clear,
3. find the gradient of \(C\) at the point where \(x = 0\)

1. (i) (a) Find, in ascending powers of \(x\), the 2nd, 3rd and 5th terms of the binomial expansion of

$$( 3 + 2 x ) ^ { 6 }$$ For a particular value of \(x\), these three terms form consecutive terms in a geometric series.
(b) Find this value of \(x\).
(ii) In a different geometric series,

• the first term is \(\sin ^ { 2 } \theta\)
• the common ratio is \(2 \cos \theta\)
• the sum to infinity is \(\frac { 8 } { 5 }\) (a) Show that

$$5 \cos ^ { 2 } \theta - 16 \cos \theta + 3 = 0$$ (b) Hence find the exact value of the 2nd term in the series.

Find the first three terms, in ascending powers of \(x\), of the binomial expansion of \(( 3 + 2 x ) ^ { 5 }\), giving each term in its simplest form.
(4)

2. (a) Find the first 3 terms, in ascending powers of \(x\), of the binomial expansion of $$( 1 + p x ) ^ { 9 }$$ where \(p\) is a constant. These first 3 terms are \(1,36 x\) and \(q x ^ { 2 }\), where \(q\) is a constant.
(b) Find the value of \(p\) and the value of \(q\).

2. (a) Find the first 4 terms, in ascending powers of \(x\), of the binomial expansion of \(( 1 - 2 x ) ^ { 5 }\). Give each term in its simplest form.
(b) If \(x\) is small, so that \(x ^ { 2 }\) and higher powers can be ignored, show that $$( 1 + x ) ( 1 - 2 x ) ^ { 5 } \approx 1 - 9 x$$ DU

Find the first 3 terms, in ascending powers of \(x\), of the binomial expansion of \(( 3 - 2 x ) ^ { 5 }\), giving each term in its simplest form.
(4)

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

$$( 3 - x ) ^ { 6 }$$ and simplify each term.