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)  $S_{63}=31815$
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 $S_k=15750$.

   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)  $S_{14}=3051757812$
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)  $S_{\mathrm{\infty }}=20480$
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)  $u_{19}=786432$;     (e)  $S_{19}=1572861$
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)   ${\displaystyle \sum _{k=1}^6{u}_k}$

   (ii)   ${\displaystyle \sum _{k=1}^{\mathrm{\infty }}{u}_k}$

   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)  $S_{\mathrm{\infty }}=\frac{243}{8}$

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