+583 If\(a(n+1)=1/2a(n)+10\)(1),and a(0)=50,find an expression for a(n) with n.
Let \(a(n+1)+k=1/2(a(n)+k)\) (2)
\(a(n+1)+k=1/2*a(n)+1/2*k\)
\(a(n+1)=1/2*a(n)-1/2*k\)
compare to the equatio (1),we find that k=-20
subsitute k=-20 into equation,we have
\(a(n+1)-20=1/2(a(n)-20)\)
or \((a(n+1)-20)/(a(n)-20)=1/2\)
\((a(n)-20)/(a(n-1)-20)=1/2\).......
I found a general equation for this with term n and b
\((a(n)-20)/(a(b)-20)={1/2}^{n-b}\)
I choose b=1,then
a(1)=1/2*a(0)+10=25+10=35
a(n)-20=(a(1)-20)(1/2)^(n-1)
a(n)=20+15*1/2^n*2=30*(1/2)^n+20
If I choose b=2,then
a(2)=1/2*a(1)+10=27.5
a(n)-20=(a(2)-20)*(1/2)^(n-2)=7.5*(1/2)^n*4=30*(1/2)^n
a(n)=20+30*(1/2)^n
Thanks for the solution,Maxwong.Any other about how to solve this problem would be wecome.
