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.

8. \(\quad \mathrm { f } ( x ) = \left( 1 + \frac { x } { k } \right) ^ { n } , \quad k , n \in \mathbb { N } , \quad n > 2\). Given that the coefficient of \(x ^ { 3 }\) is twice the coefficient of \(x ^ { 2 }\) in the binomial expansion of \(\mathrm { f } ( x )\),

1. prove that \(n = 6 k + 2\). Given also that the coefficients of \(x ^ { 4 }\) and \(x ^ { 5 }\) are equal and non-zero,
2. form another equation in \(n\) and \(k\) and hence show that \(k = 2\) and \(n = 14\). Using these values of \(k\) and \(n\),
3. expand \(\mathrm { f } ( x )\) in ascending powers of \(x\), up to and including the term in \(x ^ { 5 }\). Give each coefficient as an exact fraction in its lowest terms END

8. $$\mathrm { f } ( x ) = \left( 1 + \frac { x } { k } \right) ^ { n } , \quad k , n \in \mathbb { N } , \quad n > 2 .$$ Given that the coefficient of \(x ^ { 3 }\) is twice the coefficient of \(x ^ { 2 }\) in the binomial expansion of \(\mathrm { f } ( x )\),

1. prove that \(n = 6 k + 2\). Given also that the coefficients of \(x ^ { 4 }\) and \(x ^ { 5 }\) are equal and non-zero,
2. form another equation in \(n\) and \(k\) and hence show that \(k = 2\) and \(n = 14\). Using these values of \(k\) and \(n\),
3. expand \(\mathrm { f } ( x )\) in ascending powers of \(x\), up to and including the term in \(x ^ { 5 }\). Give each coefficient as an exact fraction in its lowest terms

12. Consider the binomial expansion of \(\left( 1 + \frac { x } { 5 } \right) ^ { n }\) in ascending powers of \(x\), where \(n\) is a positive integer.
i. Write down the first four terms of the expansion, giving the coefficients as polynomials in \(n\). The coefficients of the second, third and fourth terms of the expansion are consecutive terms of an arithmetic sequence.
ii. Show that \(n ^ { 3 } - 33 n ^ { 2 } + 182 n = 0\).
iii. Hence find the possible values of \(n\) and the corresponding values of the common difference.