Math H162 Brown Wi05

Homework 2, due Wednesday,
January 12, 2005

Use induction on these
problems.

Problem A was on the previous
homework.

** **

**Problem
B.**
(a) Show that the sum of the
first n even positive integers is

n(n
+ 1)

(b) What do you think is the sum of the
first n odd positive integers?
(Work enough examples until you can guess the answer.)

(c) Prove using induction that your answer
in (b) is correct.

**Problem
C.**
Prove that the sum of the squares of the first n positive
integers is

n(n
+ 1)(2n + 1)/6.

**Problem
D. **Prove
that** **1^{2}/[1 × 3] +** **2^{2}/[3 × 5] + É + n^{2}/[(2n Ð1)(2n + 1)] =

n(n
+ 1)/[2(2n + 1)], for all positive integers n.

**Problem
E. **Prove
that for all nonnegative integers
n, u_{n} = 2^{n}
+ 1,

Provided
that u_{0} = 2, u_{1}
= 3, and u_{ k + 1} = 3u_{ k} Ð 2u_{ k - 1} for every
positive integer k.

**Problem
F. **Prove
that if r is not 1, then for any constant a,

a
+ ar + ar^{2} + É + ar^{n - 1} = a(1 Ð r^{n})/(1
Ð r)

**Problem
G. **Prove
that the sum of the reciprocals of the square roots of the first n
positive integers is greater than the square root of n.

**Problem
H. **(a) By doing some experimental calculations
devise what you think is a correct formula for the sum of the fourth powers of
the first n positive integers. (No fair looking it
up in a book.)

(b) Prove by induction that your formula is
correct. (Chances are that if you can't prove it by induction, then your
formula is not correct.)