this is a difficult question for me please help! thankyou
A sequence is defined recursively by
an + 1 = 3an − n, a1 = 2.
Find the first six terms of the sequence.
a1 =
a2 =
a3 =
a4 =
a5 =
a6 =
To find the first six terms of the sequence, we need to use the recursive definition and keep substituting the values.
Given the recursive definition: an + 1 = 3an − n and a1 = 2, we can start by finding a2.
Substituting n=1 into the recursive equation, we get:
a2 = 3a1 - 1
Now, let's substitute the value of a1 = 2 into the equation:
a2 = 3(2) - 1
a2 = 6 - 1
a2 = 5
So, the first term of the sequence is a1 = 2, and the second term is a2 = 5.
To find a3, we use the same process:
a3 = 3a2 - 2
Substituting a2 = 5:
a3 = 3(5) - 2
a3 = 15 - 2
a3 = 13
So, the first three terms of the sequence are: a1 = 2, a2 = 5, a3 = 13.
To find a4:
a4 = 3a3 - 3
Substituting a3 = 13:
a4 = 3(13) - 3
a4 = 39 - 3
a4 = 36
Thus, the first four terms of the sequence are: a1 = 2, a2 = 5, a3 = 13, a4 = 36.
To find a5:
a5 = 3a4 - 4
Substituting a4 = 36:
a5 = 3(36) - 4
a5 = 108 - 4
a5 = 104
So, the first five terms of the sequence are: a1 = 2, a2 = 5, a3 = 13, a4 = 36, a5 = 104.
Finally, to find a6:
a6 = 3a5 - 5
Substituting a5 = 104:
a6 = 3(104) - 5
a6 = 312 - 5
a6 = 307
Hence, the first six terms of the sequence are: a1 = 2, a2 = 5, a3 = 13, a4 = 36, a5 = 104, a6 = 307.
To find the first six terms of the sequence, we will use the given recursive definition:
a(1) = 2
Now, let's find the other terms step-by-step:
a(2) = 3 * a(1) - 1
= 3 * 2 - 1
= 5
a(3) = 3 * a(2) - 2
= 3 * 5 - 2
= 15 - 2
= 13
a(4) = 3 * a(3) - 3
= 3 * 13 - 3
= 39 - 3
= 36
a(5) = 3 * a(4) - 4
= 3 * 36 - 4
= 108 - 4
= 104
a(6) = 3 * a(5) - 5
= 3 * 104 - 5
= 312 - 5
= 307
Therefore, the first six terms of the sequence are:
a1 = 2
a2 = 5
a3 = 13
a4 = 36
a5 = 104
a6 = 307