4) Let n 2 2 be an integer, such that 2" – 1 is prime number (e.g. if n = 7 then2" – 1 = 127 is a prime number). Show that then the sum of all positivedivisors of m = 2"-1.(2" – 1) is equal 2" · (2" – 1). Is this also true for allother n 2 2 (for which 2" – 1 is not prime)?

Answered: 4) Let n 2 2 be an integer, such that…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Please help find solution to this problem.
I need help with part D
2. Let, a1= 3 and for n ≥ 2, an = 2an-1 + 1, express an in terms of n. for n ≥ 3, an = 2an-1+ bn-1 for n ≥ 2 bn = bn-1 + an-1 express an in terms of n.
Which choice gives an example that supports the conjecture, and a counterexample that shows the conjecture is false? For any real number n, Vn? = n. /(9)² = 9 but V(2)² = 2 V(-9)² = 9 but V(2) = 2 V(-9)2 = 9 but V(-2)² = 2 O d. V(9)2 = 9 but V-2)2 = 2 a. b. С.
If n = 12 m (me N), prove that n "C2 "Co- - n "C4 "C6 (2+√3)² (2+√3)² (2+√3)6 +···=(−1)” 2√2 1+√3 n
9. Which of the following is equal to ()? 3.5.7..... (2n + 1) 3"n! (Зп — 1) (b) (–1)"–12·5 - 8 - .... (3n – 4) , n > 2 (a) (–1)"- 3"n! (Зп — 4) 2.5-8. 2.5.8 ..... .... (c) (–1)": (d) (–1)"- n2 2 3"n! n! (e) (–1)7-13·5 ·7. 3"n! (Зп — 2) ..... n >1
Evaluate the following 2s + 5 1 L- s2 + 6s + 34 cosh t et 5s 3t 2)2 |
4. Show that if n is a positive integer, then fo(n) (20(n) (2n) = if n is odd if n is even.
bipin
5. Prove that a natural number (am-1am-2 ·a1a0)10 is divisible by 9 if and only if the sum of the digits an-1 + an-2 +.+ ai + ao is divisible by 9. n-.
6. Let a be a number written in base 10 as x = ao + a110 +..a, 10" where 0 < a; < 10. (a) Prove that 2 divides a if and only if 2 divides ao. (b) Prove that 4 divides a if and only if 4 divides ao +2a1. (c) Prove that 8 divides a if and only if 8 divides ao + 2a1 + 4a2. (d) Prove that 5 divides a if and only if 5 divides ao. (e) Prove that 9 divides a if and only if 9 divides the sum ao + ..an of its digits. (f) Prove that 3 divides a if and only if 3 divides the sum ao + ...an of its digits. (g) Prove that 11 divides a if and only if 11 divides the alternating sum ao – a1 + az · (h) Come up with a rule for dividing by 7 and prove that it works.
Let M = 10 6 [:: -3 1 Find formulas for the entries of M", where n is a positive integer. Mn = ((-2)(7^n))+4^n 7^n-(4^n) (-2)(7^n-4^n) 7^n-(2(4^n))

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4) Let n 2 2 be an integer, such that 2" – 1 is prime number (e.g. if n = 7 then 2" – 1 = 127 is a prime number). Show that then the sum of all positive divisors of m = 2"-1.(2" – 1) is equal 2" · (2" – 1). Is this also true for all other n 2 2 (for which 2" – 1 is not prime)?