Find the first five terms of each sequence. a_(n)=2a_(n-1)-1, where a_(1)=2
The first five terms of the sequence are:
a1 = 2
a2 = 2(2) - 1 = 3
a3 = 2(3) - 1 = 5
a4 = 2(5) - 1 = 9
a5 = 2(9) - 1 = 17
is this correct
Yes, that is correct.
To find the first five terms of the sequence, we will use the given recursive formula: a_(n) = 2a_(n-1) - 1, where a_(1) = 2.
Step 1: First term (n = 1)
Using the formula with n = 1:
a_(1) = 2a_(1-1) - 1
a_(1) = 2a_(0) - 1
Since this is the initial term given, a_(1) = 2.
First term: a_(1) = 2
Step 2: Second term (n = 2)
Using the formula with n = 2:
a_(2) = 2a_(2-1) - 1
a_(2) = 2a_(1) - 1
Substituting a_(1) = 2:
a_(2) = 2(2) - 1
a_(2) = 4 - 1
a_(2) = 3
Second term: a_(2) = 3
Step 3: Third term (n = 3)
Using the formula with n = 3:
a_(3) = 2a_(3-1) - 1
a_(3) = 2a_(2) - 1
Substituting a_(2) = 3:
a_(3) = 2(3) - 1
a_(3) = 6 - 1
a_(3) = 5
Third term: a_(3) = 5
Step 4: Fourth term (n = 4)
Using the formula with n = 4:
a_(4) = 2a_(4-1) - 1
a_(4) = 2a_(3) - 1
Substituting a_(3) = 5:
a_(4) = 2(5) - 1
a_(4) = 10 - 1
a_(4) = 9
Fourth term: a_(4) = 9
Step 5: Fifth term (n = 5)
Using the formula with n = 5:
a_(5) = 2a_(5-1) - 1
a_(5) = 2a_(4) - 1
Substituting a_(4) = 9:
a_(5) = 2(9) - 1
a_(5) = 18 - 1
a_(5) = 17
Fifth term: a_(5) = 17
Therefore, the first five terms of the sequence are:
2, 3, 5, 9, 17