Coefficients in arithmetic/geometric progression

5 In the expansion of \(( a + b x ) ^ { 7 }\), where \(a\) and \(b\) are non-zero constants, the coefficients of \(x , x ^ { 2 }\) and \(x ^ { 4 }\) are the first, second and third terms respectively of a geometric progression. Find the value of \(\frac { a } { b }\).

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.

In the question you must show detailed reasoning Given that the coefficients of \(x\), \(x^2\) and \(x^4\) in the expansion of \((1 + kx)^n\) are the consecutive terms of a geometric series, where \(n \geq 4\) and \(k\) is a positive constant

1. Show that $$k = \frac{6(n-1)}{(n-2)(n-3)}$$ [4]
2. For the case when \(k = \frac{7}{2}\), find the value of \(n\). [2]
3. Given that \(k = \frac{7}{5}\), \(n\) is a positive integer, and that the first term of the geometric series is the coefficient of \(x\), find the number of terms required for the sum of the geometric series to exceed \(1.12 \times 10^{12}\). [4]

Consider the binomial expansion of \(\left(1 + \frac{x}{5}\right)^n\) in ascending powers of \(x\), where \(n\) is a positive integer.

1. Write down the first four terms of the expansion, giving the coefficients as polynomials in \(n\). [1]

The coefficients of the second, third and fourth terms of the expansion are consecutive terms of an arithmetic sequence.

1. Show that \(n^3 - 33n^2 + 182n = 0\). [3]
2. Hence find the possible values of \(n\) and the corresponding values of the common difference. [3]