Course Problems
Home
Problems
Assign Problems
Organize
Assign Problems
Add Problems
Solution Progress
TikZ Images
Compare
Difficulty
Banger Rating
PDF Management
Ctrl+S
Edit Problem
Year
Paper
Question Number
Course
-- Select Course --
LFM Pure
LFM Pure and Mechanics
LFM Stats And Pure
UFM Additional Further Pure
UFM Mechanics
UFM Pure
UFM Statistics
zNo longer examinable
Section
-- Select Section --
Coordinate Geometry
Simultaneous equations
Proof
Proof by induction
Introduction to trig
Modulus function
Matrices
Linear transformations
Invariant lines and eigenvalues and vectors
Trigonometry 2
Small angle approximation
Differentiation
Integration
Implicit equations and differentiation
Differential equations
3x3 Matrices
Exponentials and Logarithms
Arithmetic and Geometric sequences
Differentiation from first principles
Integration as Area
Vectors
Constant Acceleration
Non-constant acceleration
Newton's laws and connected particles
Pulley systems
Motion on a slope
Friction
Momentum and Collisions
Moments
Parametric equations
Projectiles
Quadratics & Inequalities
Curve Sketching
Polynomials
Binomial Theorem (positive integer n)
Functions (Transformations and Inverses)
Partial Fractions
Generalised Binomial Theorem
Complex Numbers (L8th)
Combinatorics
Measures of Location and Spread
Probability Definitions
Tree Diagrams
Principle of Inclusion/Exclusion
Independent Events
Conditional Probability
Discrete Probability Distributions
Uniform Distribution
Binomial Distribution
Geometric Distribution
Hypergeometric Distribution
Negative Binomial Distribution
Modelling and Hypothesis Testing
Hypothesis test of binomial distributions
Data representation
Continuous Probability Distributions and Random Variables
Continuous Uniform Random Variables
Geometric Probability
Normal Distribution
Approximating Binomial to Normal Distribution
Solving equations numerically
Newton-Raphson method
Sequences and Series
Number Theory
Vector Product and Surfaces
Groups
Reduction Formulae
Moments
Work, energy and Power 1
Momentum and Collisions 1
Centre of Mass 1
Circular Motion 1
Momentum and Collisions 2
Work, energy and Power 2
Centre of Mass 2
Circular Motion 2
Dimensional Analysis
Variable Force
Simple Harmonic Motion
Sequences and series, recurrence and convergence
Roots of polynomials
Polar coordinates
Conic sections
Taylor series
Hyperbolic functions
Integration using inverse trig and hyperbolic functions
Vectors
First order differential equations (integrating factor)
Complex numbers 2
Second order differential equations
Discrete Random Variables
Poisson Distribution
Approximating the Poisson to the Normal distribution
Approximating the Binomial to the Poisson distribution
Probability Generating Functions
Cumulative distribution functions
Exponential Distribution
Bivariate data
Linear regression
Moment generating functions
Linear combinations of normal random variables
Central limit theorem
Hypothesis test of a normal distribution
Hypothesis test of Pearson’s product-moment correlation coefficient
Hypothesis test of Spearman’s rank correlation coefficien
Hypothesis test of a Poisson distribution
The Gamma Distribution
Chi-squared distribution
Yates’ continuity correction
Non-parametric tests
Wilcoxon tests
Moments of inertia
Worksheet Citation (for copying)
Click the copy button or select the text to copy this citation for use in worksheets.
Problem Text
The two sequences $a_0$, $a_1$, $a_2$, $\ldots$ and $b_0$, $b_1$, $b_2$, $\ldots$ have general terms \[ a_n = \lambda^n +\mu^n \text { \ \ \ and \ \ \ } b_n = \lambda^n - \mu^n\,, \] respectively, where $\lambda = 1+\sqrt2$ and $\mu= 1-\sqrt2\,$. \begin{questionparts} \item Show that $\displaystyle \sum_{r=0}^nb_r = -\sqrt2 + \frac 1 {\sqrt2} \,a_{\low n+1}\,$, and give a corresponding result for $\displaystyle \sum_{r=0}^na_r\,$. \item Show that, if $n$ is odd, $$\sum_{m=0}^{2n}\left( \sum_{r=0}^m a_{\low r}\right) = \tfrac12 b_{n+1}^2\,,$$ and give a corresponding result when $n$ is even. \item Show that, if $n$ is even, $$\left(\sum_{r=0}^na_r\right)^{\!2} -\sum_{r=0}^n a_{\low 2r+1} =2\,,$$ and give a corresponding result when $n$ is odd. \end{questionparts}
Solution (Optional)
None
Preview
Problem
Solution
Update Problem
Cancel
Current Ratings
Difficulty Rating:
1600.0
Difficulty Comparisons:
0
Banger Rating:
1500.0
Banger Comparisons:
0
Search Problems
Press Enter to search, Escape to close