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  12/[1 × 3]  + 22/[3 × 5]  + É  + n2/[(2n Ð1)(2n + 1)] =

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

Problem E. Prove that for all nonnegative integers  n,  un = 2n + 1,

Provided that  u0 = 2, u1 = 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 + ar2 + É + arn - 1  =  a(1 Ð rn)/(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.)