The recursive sequence is defined by the formula t_n=2t_(n-1)+3tn=2tn−1+3; and t_1=-2t1=−2, how do you find t_6t6?
2 Answers
Find
Explanation:
Note that as
So we can look for a general formula for terms of the form:
t_n = 2^na + btn=2na+b
for some constants
Then we find:
2^na + b = t_n = 2t_(n-1)+3 = 2(2^(n-1)a + b) = 2^na+2b+32na+b=tn=2tn−1+3=2(2n−1a+b)=2na+2b+3
Hence we find
Then:
-2 = t_1 = 2^1 a - 3 =2a-3−2=t1=21a−3=2a−3
Hence we find
So the general formula of a term of our sequence is:
t_n = 2^(n-1)-3tn=2n−1−3
If you like, we can double check this formula:
2t_(n-1)+3 = 2(2^(n-2)-3)+3 = 2^(n-1)-6+3 = 2^(n-1)-3 = t_n2tn−1+3=2(2n−2−3)+3=2n−1−6+3=2n−1−3=tn
In particular:
t_6 = 2^5-3 = 32-3 = 29t6=25−3=32−3=29
Explanation:
Proposing
Making