University Maths Solution

Maths Question
Question 1

$\text{Evaluate the first five terms of the sequence whose }{{n}^{th}}\text{ term }{{U}_{n}}\text{ is }{{2}^{n}}+{{n}^{2}}$

Question 2

$\text{Find the formula }{{u}_{n}}\text{ for the sequence }1,-4,9,-16,25,\cdot \cdot \cdot $

Question 3

$\text{If }{{U}_{1}}=-1,\text{ }{{U}_{2}}=-5\text{ and }{{U}_{n}}=a+bn,\text{ find }a,b\text{ and }{{U}_{5}}$

Question 4

$\begin{align}  & \text{The sum of three numbers in A}\text{.P is 18 and the sum of theirs square } \\ & \text{is 206}\text{. Find the numbers} \\\end{align}$

Question 5

$\text{  Evaluate }\sum\nolimits_{r=1}^{n}{(5r-7)}$

Question 6

$\text{The product of three numbers in G}\text{.P is 1 and their sum is }-\frac{7}{3}.\text{ Find the numbers}$  

Question 7

$\text{The second term of a G}\text{.P is 24, the fifth term is 81}\text{. Find the seventh term}$ 

Question 8

$\begin{align}  & \text{The sum of the first }n\text{ terms of a geometric series is 127 and the sum of } \\ & \text{their reciprocal is }\frac{127}{64}.\text{ The first terms is 1}\text{. Find }n,\text{ and the common ratio} \\\end{align}$

Question 9

$\begin{align}  & \text{If the tenth term of a G}\text{.P is 2 and the twentieth term is }\frac{1}{512},\text{ find the first term, } \\ & \text{common ratio and sum to infinity}\text{.} \\\end{align}$

Question 10

$\begin{align}  & \text{Find the sum of the first }2n\text{ terms of the series } \\ & a+3b+2a+6b+3a+9b+\cdot \cdot \cdot  \\\end{align}$ 

Question 11

$\begin{align}  & \text{The sum of the square of three positive integers in A}\text{.P is 155}\text{. The sum of the } \\ & \text{numbers is 21}\text{. Find the numbers} \\\end{align}$

Question 12

$\begin{align}  & \text{If }a\text{ and }r\text{ are both positive, prove that the series } \\ & \log a+\log ar+\log a{{r}^{2}}+\cdot \cdot \cdot +\log a{{r}^{n-1}} \\ & \text{is an arithmetic series and find the sum of the terms} \\\end{align}$

Question 13

$\begin{align}  & \text{The sum to infinity of a G}\text{.P is }S.\text{ The sum to infinity of the cubes is }\frac{64}{13}S. \\ & \text{ Find }S\text{ and the first three terms of the }G.P \\\end{align}$

Question 14

$\text{If }p,q,r,s\text{ are any consective terms of an A}\text{.P}\text{. Show that }{{p}^{2}}-3{{q}^{2}}+3{{r}^{2}}-{{s}^{2}}=0$ 

Question 15

$\text{If }{{\alpha }^{2}}+{{\beta }^{2}},\alpha \beta +\beta \gamma \text{ and }{{\beta }^{2}}+{{\gamma }^{2}}\text{ are in G}\text{.P, Prove that }\alpha \text{,}\beta ,\gamma \text{ are also in G}\text{.P}$ 

Question 16

$\text{If }\frac{1}{\beta +\gamma },\frac{1}{\alpha +\gamma },\frac{1}{\alpha +\beta }\text{ are in A}\text{.P, Prove that }{{\alpha }^{2}},{{\beta }^{2}},{{\gamma }^{2}}\text{ are also in A}\text{.P}$ 

Question 17

$\begin{align}  & \text{Find the }{{r}^{th}}\text{ term of the series }(3\times 4)+(4\times 5)+(5\times 6)+\cdot \cdot \cdot  \\ & \text{Hence, find the sum of the first }n\text{ terms of the series} \\ & \text{Verify your answers by setting }n=3 \\\end{align}$  

Question 18

$\begin{align}  & \text{Show that }\frac{1}{r(r+2)}=\frac{1}{2}\left( \frac{1}{r}-\frac{1}{r+2} \right)\text{ and hence} \\ & \text{ Find the sum of the series }\frac{1}{1\times 3}+\frac{1}{2\times 4}+\frac{1}{3\times 5}+\cdot \cdot \cdot +\frac{1}{n(n+2)} \\\end{align}$