11. Find Un integer 21). HD w
Let n be a positive integer. For each possible pair i, j of integers with 1<i<i <n, find an n xn matrix A with the property that 1 is an eigenvalue of A with g(1) = i and a(1) = j.
8. Use mathematical induction to prove that n + + 7n 15 3 5 is an integer for all integers n > 0.
Problem 1 (**) Find a positive integer m such that φ(m) < 0.1m.
Consider a discrete-time LTI system with impulse response hn on-un-1), where jal < 1. Find the output y[n] of the system to the input x[n] = un +1].
1. Convolution of Discrete Time Signals Analytically determine the convolutions: (a) Find yn]-nun], where un] is the unit step (b) Find yln] z[n] when, where 131 <1
Say that a < 0) and k is a positive integer. Find a constant c such that tk edt < ceżat for all t > 0.
DETAILS LARCALCET7 9.8.042. ASK YOUR TEACHER Find the interval of convergence of the power series, where c> 0 and kis a positive integer. (Be sure to include a check for convergence at the endpoints of the interval. Enter your answer using Interval notation.) (-1***(x-43"
Problem 7. Let U1,U2,... be independent random variables all uniformly distributed on the unit interval, and let N be the first integer n 2 2 such that Un > Un-1. Show that for each real number 0<u < 1 !-un . 1- e-". (a) P(Ui-u and N = n) = (b) PUI S u and N is even) Problem 7. Let U1,U2,... be independent random variables all uniformly distributed on the unit interval, and let N be the first integer...
1. Let n be a positive integer with n > 1000. Prove that n is divisible by 8 if and only if the integer formed by the last three digits of n is divisible by 8.