91Ó°ÊÓ

Find the sum of the first \(n\) terms of the arithmetic progression \(2+5+8+\ldots\) Find the value of \(n\) for which the sum of the first \(2 n\) terms will exceed the sum of the first \(n\) terms by 224 . \((\mathrm{AEB}) \mathrm{p}^{\prime} 71\)

The sum to infinity of the series \(1+2 x+4 x^{2}+8 x^{3}+\ldots\), for \(-\frac{1}{2}

The first term of an arithmetic series is \(\ln x\) and the \(r\) th term is \(\ln \left(x c^{r-1}\right)\) Show that the sum \(S_{n}\) of the first \(n\) terms of the series is \(\frac{n}{2} \ln \left(x^{2} c^{n-1}\right)\). \((\mathrm{AEB}) \mathrm{p}^{\prime} 76\)

(a) By using an infinite geometric progression show that \(0.432 \mathrm{i}\), i.e. \(0.4321212121 \ldots\) is equal to \(\frac{713}{1650}\). (b) Write down the term containing \(p^{r}\) in the binomial expansion of \((q+p)^{n}\) where \(n\) is a positive integer. If \(p=\frac{1}{6}, \quad q=\frac{5}{6}\) and \(n=30\), find the value of \(r\) for which this term has the numerically greatest value. (AEB) 74

The first three terms of a geometric progression are also the first, ninth and eleventh terms, respectively, of an arithmetic progression. Given that the terms of the geometric progression are all different, find the common ratio \(r\). If the sum to infinity of the geometric progression is 8, find the first term and find the common difference of the arithmetic progression. \((J M B) p\)

Show that if \(f(r) \equiv r(r+1)(r+2)\) then $$ f(r)-f(r-1) \equiv 3 r(r+1) $$ Hence find the sum of the series $$ 2+6+12+\ldots+r(r+1)+\ldots+n(n+1) $$ \((\mathrm{U}\) of \(\mathrm{L}) \mathrm{p}\)