. Use mathematical induction to show that n Σ r=1 (2r − 1)² = 1² + 3² +5² + + (2n − 1)² = = n(2n − 1) (2n + 1) 3

Find the value of :(t +3) * [5(t – 3) – 6(t – 1)] | Select one: O z(t + 3) a(t + 2) O z(t) r(t + 1) O z(t - 1) r(t + 2) O z(t) r(t+2)

Which of the following expressions are equivalent to 2n 4n2 + 4n 10 (n+ 1) C n+ 5 4n (n+1) +. 4n (n + 1) 4n2 + 6n 5 (4n + 4n) + n(2n) 2n (4n2 + 4n) 11n + 10 D 4n (n + 1)

Consider an V4n2 + 3n – 2n for n > 1. -

(a) Find the eigenvalues and corresponding eigenvectors of the following matrix: 0-4 A = 05 -4 4 4 3 (b) The matrix A = ·( |has eigenvalues A₁ = -1 with eigenvector X₁ = [1, -2]T, and A₂ = -3 with eigenvector X₂ = [1,-1]T. 1 If P is a matrix such that P = ( X₁ X₂ ) = (-¹/₂2 - ₁1), s -1 gives a diagonal matrix with the same eigenvalues as A. -5-2 1 show that P-¹AP

8 For which values of mis x - 1 a factor of x³ + m²x² + 3mx + 1? (a) Show that n = 1 is a root of the p(n) = n³ − 8n² + 20n − 13. (b) Factor p(n). (c) For how many values of n is p(n) prime?

(Vn)(n2 +n+ 17 is prime)

Show that c12 2 dr = In (7 + 4v/3). 2 – 8x

2. Find (w + 1)(w-w+1).

2n+2 – 37 - 2. Determine Σ 6n+1 n=0