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Suppose that 3=2/x(1)=x(1)+(2/x(2))=x(2)+(2/x(3))=x(3)+(2/x(4))+...Guess an expression, in terms of n, for x(n). Then, by induction or otherwise, prove the correctness of your guess.

We have 3=2/x 1 =x 1 +2/x 2 =x 2 +2/x 3 =x 3 +2/x 4 =.... How can we guess at a general expression? One way is to try and see one by looking at the first few terms and looking for a pattern. We can calculate...
GC
Answered by George C. • STEP tutor
1554 Views

STEP 2 - 2018, Q6i): Find all pairs of positive integers (n, p), where p is a prime number, that satisfy n! + 5 = p .

The key observation here is that all numbers below n, by definition, divide n!=nx(n-1)x(n-2)x...x1.So in particular, if n greater or equal to 5, 5 divides n!, and thus is divides n!+5.Hence, if n!+5=p and n ...
MV
Answered by Marios V. • STEP tutor
2375 Views

Differentiate x^x

Write y = x^x , then we need to find dy/dx. Take the natural logarithm of both sides to give ln(y) = ln (x^x). Using log laws we can write the RHS as x ln (x). Now differentiate both sides with respect to x....
WH
Answered by William H. • STEP tutor
1583 Views

Given a differential equation (*), show that the solution curve is either a straight line or a parabola and find the equations of these curves.

This question is takes from the 2008 STEP I paper (Question 8). We are first given (dy/dx)^2 + x dy/dx + y = 0 where y is a function of x. Differentiating both sides of the equation with respect to x, gives:...
SB
Answered by Samuel B. • STEP tutor
2092 Views

Show that i^i = e^(-pi/2).

This question is very similar to differentiating x^x and the method for tackling it is quite common when powers are involved. We want to show that i^i=e^(-pi/2). Start by writing z=i^i (act like we do not kn...
JP
Answered by John P. • STEP tutor
8240 Views