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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).
Definition of Natural exponential function: The natural exponential function is f( x ) = e x where, e= 2.71828182845905........ . Hence, since e > 1 we also know that e x
Fermat's Theorem If f(x) has a relative extrema at x = c and f′(c) exists then x = c is a critical point of f(x). Actually, this will be a critical point that f′(c) =0.
We require to check the derivative thus let's use v = 60. Plugging it in (2) provides the slope of the tangent line as -1.96, or negative. Thus, for all values of v > 50 we will ha
Two tangents PA and PB are drawn to the circle with center O, such that ∠APB=120 o . Prove that OP=2AP. Ans: Given : - ∠APB = 120o Construction : -Join OP To prove : -
Q. How to divide two fractions?If you want to divide two fractions, You invert the second fraction (that means, turn it upside-down) and multiply (change the division to a
Pai is rational or irrational
HOW CAN WE TAKE SUPPOSE THE VALUES OF X AND Y
What is the slope of the line tangent to f(x)=3-2 ln(2x^2+4) at the point (4, f(4))
Explain Basic Geometric Concepts ? Points, lines, and planes are the most fundamental concepts in the study of geometry. Points A point has no length, width or heig
What is a Computer? A computer is an electronic device which senses or accepts input data, performs operations or computations on the data in a pre-arranged sequence
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