Problem 6.7. Prove √2√3 √<3 by induction.

10. (i) Prove that for all n e N, n²+2 is not divisible by 4 (ii) "Given x € R, the value of 3x-28] is greater than or equal to the value of (x-9)." State, giving a reason, if the above statement is always true, sometimes true or never true.

Problem 11. (a) The number 2 is a primitive root modulo 13. List ALL of the primitive roots modulo 13. (b) List ALL of the quadratic residues modulo 13, again taking advantage of the fact that 2 is a primitive root.

How to do this problem

24. Binomial Multiplication Patterns in Pascal's Triangle a.) What is (x+1)°? (Hint: anything to the 0 power is b.) What is (x+1)'? c.) What is (x+1)²? Caution: it is not x2+12, but is gotten by multiplying (x + 1)(x + 1). d.) Where can the coefficients and constants of each answer be found in Pascal's triangle? The coefficient is the number in front of x, and the constant is the added on number. For x + 1, the coefficient in front of the x is a 1, so we have 1x + 1. Where do you see 1 1 in the triangle? Where do you see the coefficients and constants for your answer to part c? 4 1 10 10 1 6 I 15 I 20 I 15 7 I 21 I 35 56 I 70 I 56 I 28 36 I 84 1126 | 126 I 84 I 36 1 6 1 1 35 I 21 I 7 1 8 28 8 1 9 e.) What is (x+1)³? Hint: multiply the result of part c by (x+1), then combine like terms. Explain how you know you are correct using Pascal's triangle. f.) Make a conjecture as to what (x+1)4 will equal, using Pascal's triangle. You do not have to multiply it all out to check, but instead,…

How can I evaluate 17?

Please help me solve this abstract algebra problem with a better explanation. The MI stands for mathematical induction. Thank you.

Problem 4: (4.1) Show that 2"

3. In this problem, you may use the following formulae without further justification: 2 ^²³ − ( ² - - (n(n + ¹)) ². 2 £* k k=1 n(n+1) 2 12 k=1 k² (a) Find a formula in n for each of (b) Find a formula in n for each of Σ(2k) and Σ(2k - 1). k=1 k=1 n n(n+1)(2n +1) 6 (c) Find a formula in n for each of n Σ(2k)² and Σ(2k-1)². TL k=1 TL (d) Only use the formulae above, to compute Σ(2k)³ and Σ(2k-1)³. 1³22 +33 4²+ +99³-100², which is the sum of cubes of all odd integers no greater than 100, minus the sum of squares of all even integers no greater than 100.