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The line y = 3x-4 intersects the curve y = x^2 - a, where a is an unknown constant number. Find all possible values of a.

For the line and the curve to intersect we need the for the following system of equations to have a solution. y = 3x AND y = x 2 - a The solution of the system of equations is found by solving x^2 - 3x - a =...
HK
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prove by induction that, f(n) = 2^(3n+1) + 3(5^(2n+1)) is divisible by 17 for all n>0.

With induction we start with the base case n = 1. So setting n=1 we find that f(1) = 391 which is equal to 17x23. So indeed the base case holds.We assume that for positive integers k, f(k) is divisible by 17...
MB
11427 Views

How do you solve, dy/dx=(x^2+y^2)/xy?

Substitute y/x = z then separate the variables and integrate.
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Prove by induction that 1^2 + 2^2 + 3^2 + . . . + n^2 = (1/6)n(n+1)(2n+1)

Base case - First let us consider the base case where n=1. Then the left hand side of the equation becomes 1 and the right hand side is (1/6) 1 2*3=6/6=1. Thus LHS=RHS and we are done. Induction hypothesis -...
AR
6763 Views

Prove by induction that 6^n + 4 is divisible by 5 for all integers n >= 1

As is the case with all induction questions, we first need to show that the result is true for the base case , in this case n = 1.If n = 1, we have that the LHS = 6 1 + 4 = 10, which is indeed divisible by 5...
AB
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