N - the number
14 + n = 17 |subtract 14 from both sides
n = 3
Sample Space = {HH,HT,TH,TT} - Hope this helps!
16, first you need to add on the four he lost to the 18 he ended with.
<u>Answer:</u>
The correct answer option is A. 
.
<u>Step-by-step explanation:</u>
We are given the following geometric sequence and we are to find its 8th term:
Here 
and common ratio 
.
The formula we will use to find the 8th term is:
nth term = 
Substituting the values in the formula to get:
8th term = 
8th term = 
Hi there what you need is lagrange multipliers for constrained minimisation. It works like this,
V(X)=α2σ2X¯1+β2\sigma2X¯2
Now we want to minimise this subject to α+β=1 or α−β−1=0.
We proceed by writing a function of alpha and beta (the paramters you want to change to minimse the variance of X, but we also introduce another parameter that multiplies the sum to zero constraint. Thus we want to minimise
f(α,β,λ)=α2σ2X¯1+β2σ2X¯2+λ(\alpha−β−1).
We partially differentiate this function w.r.t each parameter and set each partial derivative equal to zero. This gives;
∂f∂α=2ασ2X¯1+λ=0
∂f∂β=2βσ2X¯2+λ=0
∂f∂λ=α+β−1=0
Setting the first two partial derivatives equal we get
α=βσ2X¯2σ2X¯1
Substituting 1−α into this expression for beta and re-arranging for alpha gives the result for alpha. Repeating the same steps but isolating beta gives the beta result.
Lagrange multipliers and constrained minimisation crop up often in stats problems. I hope this helps!And gosh that was a lot to type!xd
