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1. Suppose n ≡ 7 (mod 8). Show that n ≠ x2 + y2 + z2 for any x, y, z ε Z.
2. Prove ∀n ε Z, that n is divisible by 9 if and only if the sum of its digits is divisible by 9.
3. Prove that it is always possible to make postage of exactly n cents for all n ≥ 32 using only 5 and 9 cent stamps.
4. Prove that every fourth Fibonacci number is a multiple of 3.
In other words, show that 3 | f4n ∀n ≥ 1.
5. Let bn be the sequence recursively defined by b0 = 1, b1 = 5 and, for n > 1,
bn = b[n/3]+2b[n/3]
(a) Compute b26 and b27.
(b) Guess a formula for bn when n = 3t for t ≥ 0 and then use mathematical induction to prove that your guess is correct. (Be sure to include a careful statement of what you are trying to prove).
What is algebra?
Is this given matrices are called equal Matrices?
8...
5 2 ----- - ----- x-1 x+1 -------------------- x 1 ----- + ----- x-1 x+1
Classify the following discrete-time signals as energy or power signals. If the signal is of energy type, find its energy. Otherwise, find the average power of the signal. X 1
how do you solve 45 = n/5 = 5
use an expression to write an expression with five 3s that has a value of 0
Find out the length of Hamiltonian Path in a connected graph of n vertices. Ans: The length of Hamiltonian Path in a connected graph of n vertices is n-1.
If α & ß are the zeroes of the polynomial 2x 2 - 4x + 5, then find the value of a.α 2 + ß 2 b. 1/ α + 1/ ß c. (α - ß) 2 d. 1/α 2 + 1/ß 2 e. α 3 + ß 3 (Ans:-1, 4/5 ,-6,
find any integer from 1-128 on a logarithmic scale
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