Review exercises  —  sequences and series

1. An arithmetic sequence has the terms: \(u_1=27\) and \(u_2=39\).
   

   (a)   Write down the common difference.

   (b)   Write down the \(10^{th}\) term.

   (c)   Write down the \(n^{th}\) term.

   (d)   Calculate the sum of the first 10 terms.

   Solutions:    (a)  \(d=12\);     (b)  \(u_{10}=135\);     (c)  \(u_n=15+12n\);     (d)  \(S_{10}=810\)
2. In an arithmetic sequence \(u_1=1001\) and \(u_{15}=777\).
   

   (a)   Write down the \(n^{th}\) term.

   (b)   Find the number of positive terms.

   (c)   Write down the smallest positive term.

   (d)   Calculate the sum of all positive terms.

   Solutions:    (a)  \(u_n=1017-16n\);     (b)  63 positive terms;     (c)  \(u_{63}=9\);     (d)  \(S_{63}=31\,815\)
3. Numbers \(u_1=\sqrt{m},~ u_2=m-12\) and \(u_3=m+\sqrt{m}\) are the first three terms of an arithmetic sequence.
   

   (a)   Find \(m\).

   (b)   Write down the general term.

   (c)   Given that the \(n^{th}\) term of this sequence is 600, find \(n\).

   The sum of the first \(k\) terms is labelled \(S_k\).

   (d)   Find \(k\) so that \(S_k=15\,750\).

   Solutions:    (a)  \(m=36\);     (b)  \(u_n=18n-12\);     (c)  \(n=34\);     (d)  \(k=42\)
4. In a geometric sequence \(u_1=2\) and \(u_2=10\).
   

   (a)   Find the common ratio.

   (b)   Write down the \(n^{th}\) term.

   The first \(n\) terms of this sequence are less than \(10^{10}\).

   (c)   Find \(n\).

   (d)   Find the sum of the first \(n\) terms.

   Solutions:    (a)  \(r=5\);     (b)  \(u_n=2\cdot 5^{n-1}\);     (c)  \(n=14\);     (d)  \(S_{14}=3\,051\,757\,812\)
5. In an infinite geometric sequence \(u_1=5120\) and \(u_6=1215\).
   

   (a)   Find the common ratio.

   (b)   Find the terms \(u_2,~ u_3,~ u_4\) and \(u_5\).

   (c)   Find the sum of the infinite sequence.

   Solutions:    (a)  \(r=\frac{3}{4}\);     (b)  \(u_2=3840,~ u_3=2880,~ u_4=2160,~ u_5=1620\);     (c)  \(S_{\infty}=20480\)
6. Numbers \(u_1=m,~ u_2=m+3\) and \(u_3=m^2-2m+9\) are the first three terms of an infinite geometric sequence.
   

   (a)   Find \(m\).

   (b)   Write down the \(n\)-th term.

   Some terms in this sequence have the value less than 1 million.

   (c)   Find how many terms have the value less than 1 million.

   (d)   Find the lagest among these terms.

   (e)   Calculate the sum of these terms.

   Solutions:    (a)  \(m=3\);     (b)  \(u_n=3\cdot 2^{n-1}\);     (c)  19 terms;     (d)  \(u_{19}=786\,432\);     (e)  \(S_{19}=1\,572\,861\)
7. The \(n\)-th term of a sequence is \(u_n=\frac{81}{8}\cdot\left(\frac{2}{3}\right)^{n-1}\).
   

   (a)   Write down terms \(u_4\) and \(u_5\).

   (b)   Show that this is a geometric sequence.

   (c)   Calculate the following two sums, giving your results in exact form:

   (i)   \({\displaystyle \sum\limits_{k=1}^6 u_k}\)

   (ii)   \({\displaystyle \sum\limits_{k=1}^{\infty} u_k}\)

   Solutions:    (a)  \(u_4=3,~ u_5=2\);     (b)  \(\frac{u_{n+1}}{u_n}=\frac{2}{3}=\mathrm{constant}\);     (c)(i)  \(S_6=\frac{665}{24}\);     (ii)  \(S_{\infty}=\frac{243}{8}\)

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