Three consecutive terms of an arithmetic sequence are \(x-3 \ , \ 12 \ ,\ 3x-5\)

Find \(x\)

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The 2nd term of an arithmetic sequence is 19 and the 5th term is 37.

a) Find the 10th term

b) The sum of the first n terms of this sequence exceeds 1000. Find the least value of n

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Find the sum of all the integers between 100 and 1000 that are divisible by 9

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An arithmetic sequence has first term U₁ and common difference d. The sum of the first 17 terms is 136.

a) Show that \(U_1+8d=8\)

The sum of the 2nd and the 3rd terms is 42.

b) Find d.

The nth term of the sequence is U_n.

c) Find the value of \(\sum _{ r=4 }^{ 17 }{ { U }_{ n } } \)

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In an arithmetic sequence, the 9th term is 4 times the 5th term. The sum of the first 2 terms is -13.

Find the 10th term

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The first terms of a sequence are log₃ x , log₃ x² , log₃ x³ , ...

Find x if the sum of the first 9 terms is 135

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Three terms of a geometric sequence are \(x+6 \ , \ 12 \ ,\ x-1\)

Find the possible values of x

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Consider the geometric sequence where the first term is 45 and the second term is 36.

a) Find the least value of n such that the nth term of the sequence is less than 1

b) Find the least value of n such that the sum of the first n terms of the sequence is more than 200.

c) Find the sum to infinity.

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The sum to infinity of a geometric series is 27.

The sum of the first 3 terms is 19.

Find the common ratio

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The 2nd, 3rd and 6th terms of an arithmetic sequence with common difference \(d, \ d\neq 0\) form the first three terms of a geometric sequence, with common ratio, r.

The 1st term of the arithmetic sequence is -2.

a) Find d.

The sum of the first n terms of the geometric sequence exceeds the sum of the first n terms of the arithmetic sequence by at least 1000.

b) Find the least value of n for which this occurs.

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a) Jessica takes out a loan of $200 000 to buy an appartment. The interest rate is 4% and is calculated at the end of each year. Calculate to the nearest dollar the amount Jessica would owe the bank after 15 years.

b) In order to pay of the loan, she pays $P into a bank at the end of each year. She receives an interest rate of 2.5% per year. Find the amount saved after 15 years.

c) What must be the value of P so that she has saved enough money to pay off the loan.

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a. Given that 3^a =27 , write down the value of a

b. Hence of otherwise solve 27^x+4 = 9^3x+1

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Solve the equation log₂(x - 3) = 1 - log₂(x - 4)

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Solve the equation \(3^{ x-1 }=\frac { 2 }{ { 4 }^{ x+1 } } \) , giving your answer in the form \(x=\frac{lna}{lnb}\) , where a and b are rational numbers.

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\(a=\log _{ 2 }{ 2+\log _{ 2 }{ \frac { 3 }{ 2 } + } } \log _{ 2 }{ \frac { 4 }{ 3 } + } \quad ...\quad +\log _{ 2 }{ \frac { 32 }{ 31 } } \)

Given that \(a\in \mathbb{Z}\) , find the value of a

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The first three terms of a geometric sequence are \(\log _{ 3 }{x\quad ,\quad \log _{ 9 }{ x\quad ,\quad \log _{ 81 }{ x } } } \)

Find the value of x if the sum to infinity is 8.

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A five-digit number is formed by using the digits 1-5 exactly once.

1. How many five-digit numbers are there?
2. How many of these five-digit numbers are even?

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a) Expand (x - 3)⁴ and simplify your result

b) Hence find the x³ term in (x - 2)(x - 3)⁴ .

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Find the term independent of x in the expansion \((2x- \frac{3}{x^2})^6\)

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a) Use the binomial theorem to expand \(\sqrt{4-9x}\) in ascending powers of x up to and including x³

b) State the value of x for which the expansion is valid.

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The x term in the expansion \((4+2x)^3(2+ax)^4\) is -4608x

Find a

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In the expansion \((1+bx)^n\), the coefficient of the x term is -6 and the coefficient of the x² term is 27.

Work out b and n.

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a) Write \(f(x)=\frac{2+x}{1+x-2x^2}\) as the sum of partial fractions

b) Find the binomial expansion of f(x) in ascending powers of x up to and including x²

c) State the values of x for which the series is valid

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​\(\large(\textbf{a}+2x)^3(4-x)^4 = 6912 + \textbf{b}x +...\)​

Find a and b

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a) Verify that \(\frac{1}{x^2}-\frac{1}{\left(x+1\right)^2}=\frac{2x+1}{x^2\left(x+1\right)^2} \) for x = -2

b) Prove that \(\frac{1}{x^2}-\frac{1}{\left(x+1\right)^2}\equiv\frac{2x+1}{x^2\left(x+1\right)^2} \) for all x

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Prove that the sum of three consecutive integers is divisible by 3

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a) Verify that x² - 4x + 5 is positive for x = -1

b) Prove that x² - 4x + 5 is positive for all x

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Prove that the difference between the square of any two consecutive odd integers is divisible by 8

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a) Verify that \(^3 C_1\ +\ ^3 C_2\ =\ ^4C_2\)

b) Prove that \(^{n-1} C_{r-1}\ +\ ^{n-1} C_r\ =\ ^nC_r\)

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c) Hence, use proof by contradiction to prove that \(log_25\) is irrational

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Prove by contradiction that a rational number + an irrational number = irrational number.

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Prove that \(\sqrt{3}\) is irrational

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Prove that \(\sqrt[3]{5}\) is irrational

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Prove by contradiction that the length of the hypotenuse of a right-angled triangle is less than the sum of the other two sides.

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Prove that \(\sqrt[n]{p}\) is irrational, given that p is prime.

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Prove by contradiction that there are no rational roots to the equation \(x^3+x+1=0\)

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Prove by induction that \(\sum _{ r=1 }^{ n }{(r\times{ 2 }^{ r-1 })} =(n-1)2^ n+1 \ ,\ n\in\mathbb{Z^+}\)

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Let \(y = sinx\)

Prove by induction that \(\frac{d^ny}{dx^n}=sin(x+\frac{n\pi}{2})\)

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Prove by induction that \(sinx+sin3x+sin5x+...+sin(2n-1)x=\frac{1-cos2nx}{2sinx} , \quad n\in\mathbb{Z^+} ,\quad sinx\neq 0\)

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It is given that \(z_1=2+3i\) and \(z_2=4+ai\)

Find a if \(Im(z_1z_2^*)=0\)

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Let \(z=a+bi\)

Find a and b if \(z^2=\left|z\right|^2-4\)

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Show that \((1-\frac{1}{3} e^{2iθ})(1-\frac{1}{3} e^{-2iθ})=\frac{1}{9}(10-6\cos 2θ)\)

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Given that z is a complex number and \(\frac{3z-4}{5}=\frac{p-2i}{3-i}\), where \(p \in\mathbb{R} \)

a) Express z in the form \(a+bi\), \(a,b\in\mathbb{R}\)

Given that \(arg(z)=-\frac{\pi}{2}\)

b) Find the value of p

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a) Given that \(z=a+bi\), show that \(z+\frac{1}{z}=2cos\theta\)

b) Hence, show that \(z^n+\frac{1}{z^n}=2cosn\theta\)

c) Use the Binomial expansion to expand \((z+{1\over z})^5\)

d) Hence, show that \(cos^5 θ={1 \over 16}cos⁡5θ+{5 \over 16} cos3θ+{5\over 8} cosθ\)

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\(z=-2+2\sqrt{3}i\)

a) Find |z| and arg(z)

b) Find \(z^6\) and simplify your answer

c) Given that \(w^4=z^3\) , find the values of \(w\) giving your answers in the form \(a+bi\)

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Find the values of n such that \((\sqrt{3}-i)^n\) is a real number

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a) By writing \(\frac{\pi}{12}=\frac{\pi}{3}-\frac{\pi}{4}\), show that \(sin(\frac{\pi}{12})=\frac{\sqrt{6}}{4}-\frac{\sqrt{2}}{4}\)

b) Work out \(cos(\frac{\pi}{12})\)

c) Hence, find the roots of the equation \(z^4=2+2\sqrt{3}i\), giving answers in the form \(z=a+ib\)

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a) Find the roots of the equation \(z^4-1=0\)

b) Find the roots of the equation \(z^4+1=0\)

c) Show that roots of \(z^4-1=0\) and \(z^4+1=0\) together make the roots of \(z^8-1=0\)

d) Hence, find all the roots to \(z^6+z^4+z^2+1=0\)

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\(\frac{2}{1+i}\) is a root to the quadratic equation z² + px + q = 0

b) Hence find the values of p and q.

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2 - 3i is a root of the equation \(z^3-7z^2+az+b=0\quad ,a, b\in\ \mathbb{R}\)

Work out a and b and the other roots of the equation.

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The quartic equation \(z^4+az^3+bz^2+cz+d\) has roots 2 + i and 2i

b) Find the values of a , b , c and d

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The equation \(2z^{ 4 }−9z^{ 3 }+pz^{ 2 }+qz−174=0 \quad,\quad p,q\in\mathbb{Z}\) has two real roots \(\alpha\) and \(\beta\) and two complex roots \(\gamma\) and \(\delta\) where \(\gamma=2-5i\).

a. Show that \(\alpha+\beta=\frac{1}{2}.\)

b. Find \(\alpha\beta\).

c. Hence find the two real roots α and β.

d. Find the values of p and q.

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