Homework Answers
The Given problem is solved using a process called Mathematical Induction.
The formulas used to solve the given problem is 1,n,n+1.
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1. Suppose that A=1001 00 1 where a is a real number. Prove that 1 1...
Suppose a is a real number and 1 + a > 0. Prove that (1 +...
Suppose a is a real number and 1 + a > 0. Prove that (1 + a)" > 1+ na for every integer n > 1.
a be a real number . If a--a, prove that either a 0 or a 1....
a be a real number . If a--a, prove that either a 0 or a 1. 8. (Pigeonhole Principle) Suppose we place m pigeons in n pigeonholes, where m and n are positive integers. If m > n, show that at least two pigeons must be placed in the same pigeonhole. [Hint (from Robert Lindahl of Morehead State University): For i 1, 2, . . . , n, let Xi denote the number of pigeons that are placed in the...
. 1. Prove by induction that for all integers n≥1, 4+8+12+...+4n = 2n^2+2n 2. A number...
. 1. Prove by induction that for all integers n≥1, 4+8+12+...+4n = 2n^2+2n 2. A number a is divisible by b if the remainder of dividing a by b is zero. For example 10 is divisible by 5 but 11 is not divisible by 5. Prove by induction that for all integers n≥1,11^n - 6 is divisible by 5. 3. Prove by induction that for all integers n ≥ 1, 3^n ≥ 2^n+n^2
+00 bn are series with positive terms an and (a) Suppose that O bn is convergent. Prove that if "...
+00 bn are series with positive terms an and (a) Suppose that O bn is convergent. Prove that if "limo and that I1 n= 1 +00 then Σ an is also convergent. (b) Use part (a) to show that the series converges t In (n)
+00 bn are series with positive terms an and (a) Suppose that O bn is convergent. Prove that if "limo and that I1 n= 1 +00 then Σ an is also convergent. (b) Use part...
Prove using mathematical induction that for every positive integer n, = 1/i(i+1) = n/n+1. 2) Suppose...
Prove using mathematical induction that for every positive integer n, = 1/i(i+1) = n/n+1. 2) Suppose r is a real number other than 1. Prove using mathematical induction that for every nonnegative integer n, = 1-r^n+1/1-r. 3) Prove using mathematical induction that for every nonnegative integer n, 1 + i+i! = (n+1)!. 4) Prove using mathematical induction that for every integer n>4, n!>2^n. 5) Prove using mathematical induction that for every positive integer n, 7 + 5 + 3 +.......
Suppose that A is diagonalizable and all eigenvalues of A are positive real numbers. Prove that...
Suppose that A is diagonalizable and all eigenvalues of A are
positive real numbers. Prove that det (A) > 0.
(1 point) Suppose that A is diagonalizable and all eigenvalues of A are positive real numbers. Prove that det(A) > 0. Proof: , where the diagonal entries of the diagonal matrix D are Because A is diagonalizable, there is an invertible matrix P such that eigenvalues 11, 12,...,n of A. Since = det(A), and 11 > 0,..., n > 0,...
6.2 Prove that for fixed positive integers k and n, the number of partitions of n...
6.2 Prove that for fixed positive integers k and n, the number of partitions of n is equal to the number of partitions of 2n + k into n + k parts.
5. Let {xn} and {yn} be sequences of real numbers such that x1 = 2 and...
5. Let {xn} and {yn} be sequences of real numbers such that x1 =
2 and y1 = 8 and for n = 1,2,3,···
x2nyn + xnyn2 x2n + yn2 xn+1 = x2 + y2 and yn+1 = x + y .
nn nn
(a) Prove that xn+1 − yn+1 = −(x3n − yn3 )(xn − yn) for all
positive integers n.
(xn +yn)(x2n +yn2) (b) Show that 0 < xn ≤ yn for all positive
integers n.
Hence, prove...
1. (Integers: primes, divisibility, parity.) (a) Let n be a positive integer. Prove that two numbers...
1. (Integers: primes, divisibility, parity.) (a) Let n be a positive integer. Prove that two numbers na +3n+6 and n2 + 2n +7 cannot be prime at the same time. (b) Find 15261527863698656776712345678%5 without using a calculator. (c) Let a be an integer number. Suppose a%2 = 1. Find all possible values of (4a +1)%6. 2. (Integers: %, =) (a) Suppose a, b, n are integer numbers and n > 0. Prove that (a+b)%n = (a%n +B%n)%n. (b) Let a,...
Problem 8: (i) Use the Principle of Mathematical Induction to prove that 2n+1(-1)" + 1 1...
Problem 8: (i) Use the Principle of Mathematical Induction to prove that 2n+1(-1)" + 1 1 – 2 + 22 – 23 + ... + (-1)22" = for all positive integers n. (ii) Use the Principle of Mathematical Induction to prove that np > n2 + 3 for all n > 2.
Suppose a is a real number and 1 + a > 0. Prove that (1 + a)" > 1+ na for every integer n > 1.
a be a real number . If a--a, prove that either a 0 or a 1. 8. (Pigeonhole Principle) Suppose we place m pigeons in n pigeonholes, where m and n are positive integers. If m > n, show that at least two pigeons must be placed in the same pigeonhole. [Hint (from Robert Lindahl of Morehead State University): For i 1, 2, . . . , n, let Xi denote the number of pigeons that are placed in the...
+00 bn are series with positive terms an and (a) Suppose that O bn is convergent. Prove that if "limo and that I1 n= 1 +00 then Σ an is also convergent. (b) Use part (a) to show that the series converges t In (n)
+00 bn are series with positive terms an and (a) Suppose that O bn is convergent. Prove that if "limo and that I1 n= 1 +00 then Σ an is also convergent. (b) Use part...
Prove using mathematical induction that for every positive integer n, = 1/i(i+1) = n/n+1. 2) Suppose r is a real number other than 1. Prove using mathematical induction that for every nonnegative integer n, = 1-r^n+1/1-r. 3) Prove using mathematical induction that for every nonnegative integer n, 1 + i+i! = (n+1)!. 4) Prove using mathematical induction that for every integer n>4, n!>2^n. 5) Prove using mathematical induction that for every positive integer n, 7 + 5 + 3 +.......
Suppose that A is diagonalizable and all eigenvalues of A are
positive real numbers. Prove that det (A) > 0.
(1 point) Suppose that A is diagonalizable and all eigenvalues of A are positive real numbers. Prove that det(A) > 0. Proof: , where the diagonal entries of the diagonal matrix D are Because A is diagonalizable, there is an invertible matrix P such that eigenvalues 11, 12,...,n of A. Since = det(A), and 11 > 0,..., n > 0,...
6.2 Prove that for fixed positive integers k and n, the number of partitions of n is equal to the number of partitions of 2n + k into n + k parts.
5. Let {xn} and {yn} be sequences of real numbers such that x1 =
2 and y1 = 8 and for n = 1,2,3,···
x2nyn + xnyn2 x2n + yn2 xn+1 = x2 + y2 and yn+1 = x + y .
nn nn
(a) Prove that xn+1 − yn+1 = −(x3n − yn3 )(xn − yn) for all
positive integers n.
(xn +yn)(x2n +yn2) (b) Show that 0 < xn ≤ yn for all positive
integers n.
Hence, prove...
1. (Integers: primes, divisibility, parity.) (a) Let n be a positive integer. Prove that two numbers na +3n+6 and n2 + 2n +7 cannot be prime at the same time. (b) Find 15261527863698656776712345678%5 without using a calculator. (c) Let a be an integer number. Suppose a%2 = 1. Find all possible values of (4a +1)%6. 2. (Integers: %, =) (a) Suppose a, b, n are integer numbers and n > 0. Prove that (a+b)%n = (a%n +B%n)%n. (b) Let a,...
Problem 8: (i) Use the Principle of Mathematical Induction to prove that 2n+1(-1)" + 1 1 – 2 + 22 – 23 + ... + (-1)22" = for all positive integers n. (ii) Use the Principle of Mathematical Induction to prove that np > n2 + 3 for all n > 2.
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