Advanced Math/AS level maths- Sequences and Logs
Hi here are the questions:
1) Given that log(2)y=log(4)2, find the value of y.
2) A sequence of terms is defined by Un+1=kUn+3. U1=15 and U2=6. Find the value of the constant k. (I've got the answer to this which is 1/5). The next part of the question is: given that Un converges to a limiting value L, find an equation for L and hence find the value of L. (Really stuck on this part :/)
For the logs question please could you explain it to me in steps because I'm hopeless at this particular topic. Much appreciated! Thanks
1) You need to first understand the definition of logarithm. If the logarithm of the number Z to base p equals q, written as
log(p) Z = q
it then simply means that
Z = p^q (read as Z equals p raised to power of q)
Now, let log(2) y = x, then we know that y = 2^x
But also from the equation, log(4) 2 = x, which means that
2 = 4^x
2^1 = (2^2)^x
2^1 = 2^(2x)
1 = 2x
x = 1/2
and so y = 2^(1/2) = √2
2) The meaning of U(n) converging to a limiting value is that as n approaches infinity (n→∞), the subsequent values of U stays the same for these large values of n. If you think about it, this just means that in the range of these values, U(n+1) = U(n) = L.
But the recurrence relation was given by U(n+1) = (1/5)U(n) + 3, so
L = (1/5)L + 3
(4/5)L = 3
L = 3(5/4) = 15/4
Let me know if anything is unclear.