Serveral mathematics questions

1.Given that x + 1 / x = n and n > 2, express the following in terms of n.
(a) x – 1 / x
x + 1 / x = n
( x + 1 / x )2 = n2
x2 + 2 x ( 1 / x ) + ( 1 / x )2 = n2
x2 + 2 + 1 / x2 = n2
x2 + 1 / x2 = n2 – 2
x2 – 2 + 1 / x2 = n2 – 4
( x – 1 / x )2 = n2 – 4
x – 1 / x =√( n2 – 4 )


(b) x^2 – 1 / ( x^2 )
x2 – 1 / x2
= x2 – ( 1 / x )2
= ( x + 1 / x ) ( x – 1 / x )
= n√( n2 – 4 )


2. Tommy deposits $P in a bank on the first day of 2005, and on the first day of each successive year, he will deposit an amount 20% more than the previous year's deposit. It is given that the interest rate is 4% p.a., compounded yearly.

(a) Find the total amount, in terms of P, that Tommy will receive at the end of
(i) 2005,
$P ( 1 + 4% )

(ii) 2006,
[ $P ( 1 + 4% ) + $P ( 1 + 20% ) ] ( 1 + 4 % )
= $P ( 1 + 4% )2 + $P ( 1 + 20% ) ( 1 + 4 % )

(iii) 2007.
[ $P ( 1 + 4% )2 + $P ( 1 + 20% ) ( 1 + 4 % ) + $P ( 1 + 20% )2 ] ( 1 + 4% )
= [ $P ( 1 + 4% )3 + $P ( 1 + 20% ) ( 1 + 4 % )2 + $P ( 1 + 20% )2 ( 1 + 4% )

(b) Using the formula:
a+aR+aR^2+aR^3+...+aR^(n-1) = [a(1-R^n)]/(1-R)
and the result of (a), prove that at the end of 2024, Tommy will receive a total amount of $6.5P(1.2^20-1.04^20).
Tommy will receive at the end of 2024 :
$P ( 1 + 4% )20 + $P ( 1 + 20% ) ( 1 + 4 % )19 + $P ( 1 + 20% )2 ( 1 + 4% )18 + . . . + $P ( 1 + 20% )18 ( 1 + 4 % )2 + $P ( 1 + 20% )19 ( 1 + 4% )
= $P ( 1 + 4% )20 { 1 – [ (1 + 20% ) / ( 1 + 4% ) ]20 } / [ 1 – ( 1 + 20% ) / ( 1 + 4% ) ]
= $P ( 1.04 )20 [ 1 – ( 1.2 )20 / ( 1.04 )20 ] / ( 1 - 1.2 / 1.04 )
= $P ( 1.2 20 – 1.0420 ) / ( 0.16 / 1.04 )
= $P ( 1.2 20 – 1.0420 ) (1.04 / 0.16 )
= $ 6.5 P ( 1.2 20 – 1.0420 )

(c) If P=4500, will Tommy receive a total amount of more than one million dollars at the end of 2024? Explain your answer.
Tommy will receive at the end of 2024 :
$ 6.5 X 4500 X ( 1.2 20 – 1.0420 ) = $1 057 284 > $1 000 000
Tommy will receive a total amount of more than one million dollars at the end of 2024.