- 1.2.1: Provethat 1/1 .2+ 1/2.3 + ... + l/n(n + 1) = n/(n + 1) for all n EN.
- 1.2.2: Provethat 13+ 23+... + n3 = [4n(n + 1)]2for all n EN.
- 1.2.3: Provethat 3 + 11+ . . . + (8n - 5) = 4n2 - n for all n E N.
- 1.2.4: Provethat12+ 32+... + (2n- 1)2= (4n3- n)/3 for alln EN.
- 1.2.5: Provethat 12- 22+ 32+... + (_l)n+1n2 = (_1)n+1n(n+ 1)/2 for alln EN.
- 1.2.6: Prove that n3 + 5n is divisible by 6 for all n EN.
- 1.2.7: Prove that 52n- 1 is divisible by 8 for illn EN.
- 1.2.8: Prove that 5n - 4n - 1 is divisible by 16 for all n EN.
- 1.2.9: Prove that n3 + (n + 1)3 + (n + 2)3 is divisible by 9 for all n E N. 1
- 1.2.10: Conjecture a fonnula for the sum III .3+ 1/3 .5+ . . . + 1/(2n - 1)...
- 1.2.11: Conjecture a fonnula for the sum of the first n odd natural numbers...
- 1.2.12: Prove the Principle of Mathematical Induction 1.2.3 (second version...
- 1.2.13: Prove that n < 2n for all n EN. 1
- 1.2.14: Prove that 2n < n! for all n 2: 4, n E N.
- 1.2.15: Prove that 2n - 3 ::: 2n-2 for all n 2: 5, n EN.
- 1.2.16: Find all natural numbers n such that n2 < 2n. Prove your assertion.
- 1.2.17: Find the largest natural number m such that n3 - n is divisible by ...
- 1.2.18: Prove that IIv'! + 1/../2 + + 1/..,1n> ..,Infor all n EN. 1
- 1.2.19: Let S be a subset of N such that (a) 2k E S for all kEN, and (b) if...
- 1.2.20: Let the numbers xn be defined as follows: xl := 1, x2 := 2, and xn+...
Solutions for Chapter 1.2: Mathematical Induction
Full solutions for Introduction to Real Analysis | 3rd Edition
Speed of rotation, typically measured in radians or revolutions per unit time
artesian coordinate system
An association between the points in a plane and ordered pairs of real numbers; or an association between the points in three-dimensional space and ordered triples of real numbers
A circular graphical display of categorical data
See Geometric sequence.
Decreasing on an interval
A function f is decreasing on an interval I if, for any two points in I, a positive change in x results in a negative change in ƒ(x)
A set of points in a plane, the absolute value of the difference of whose distances from two fixed points (the foci) is a constant.
The difference between the third quartile and the first quartile.
Jump discontinuity at x a
limx:a - ƒ1x2 and limx:a + ƒ1x2 exist but are not equal
Length of a vector
See Magnitude of a vector.
Numerical derivative of ƒ at a
NDER f(a) = ƒ1a + 0.0012 - ƒ1a - 0.00120.002
The graph of a quadratic function, or the set of points in a plane that are equidistant from a fixed point (the focus) and a fixed line (the directrix).
In an experimental study, an inactive treatment that is equivalent to the active treatment in every respect except for the factor about which an inference is to be made. Subjects in a blind experiment do not know if they have been given the active treatment or the placebo.
Range (in statistics)
The difference between the greatest and least values in a data set.
Function of the form ƒ(x)/g(x) where ƒ(x) and g(x) are polynomials and g(x) is not the zero polynomial.
Solve a triangle
To find one or more unknown sides or angles of a triangle
Spiral of Archimedes
The graph of the polar curve.
Standard unit vectors
In the plane i = <1, 0> and j = <0,1>; in space i = <1,0,0>, j = <0,1,0> k = <0,0,1>
The function y = tan x
The points x, y, 0 in Cartesian space.
The scale of the tick marks on the y-axis in a viewing window.