Find the first five terms of the given recursively defined sequence.
an = 3an − 1 + 6 and a1 = 1
a1=
a2=
a3=
a4=
a5=
I am sure you meant:
an = 3a(n-1) + 6
in this notation , given any an
an-1 would be the previous term.
so if a1 = 1
then a2 = 3a1 + 6
= 3(1) + 6 = 9
a3 = 3a2 + 6
= 3(9) + 6 = 33
continue building them that way.
how did you get 9? on a3?
a1 = 1
To find a2, plug in n = 2 into the equation: a2 = 3a2-1 + 6
a2 = 3a1 + 6 = 3(1) + 6 = 3 + 6 = 9
To find a3, plug in n = 3 into the equation: a3 = 3a3-1 + 6
a3 = 3a2 + 6 = 3(9) + 6 = 27 + 6 = 33
To find a4, plug in n = 4 into the equation: a4 = 3a4-1 + 6
a4 = 3a3 + 6 = 3(33) + 6 = 99 + 6 = 105
To find a5, plug in n = 5 into the equation: a5 = 3a5-1 + 6
a5 = 3a4 + 6 = 3(105) + 6 = 315 + 6 = 321
Therefore, the first five terms of the sequence are:
a1 = 1
a2 = 9
a3 = 33
a4 = 105
a5 = 321
To find the first five terms of the given recursively defined sequence, we can use the formula for the nth term of a recursively defined sequence.
Given the recursive formula: an = 3an − 1 + 6
We also know the initial term a1 = 1.
To find the second term (a2), we substitute n = 2 into the recursive formula:
a2 = 3a2 − 1 + 6
Since we don't know the value of a2, we need to find the first term (a1) to substitute it into the equation.
We already know a1 = 1, so substituting it into the equation:
a2 = 3(1) − 1 + 6
a2 = 3 − 1 + 6
a2 = 8
So, the second term of the sequence (a2) is 8.
To find the third term (a3), we substitute n = 3 into the recursive formula:
a3 = 3a3 − 1 + 6
Substituting in the values of previous terms:
a3 = 3(8) − 1 + 6
a3 = 24 − 1 + 6
a3 = 29
So, the third term of the sequence (a3) is 29.
To find the fourth term (a4), we substitute n = 4 into the recursive formula:
a4 = 3a4 − 1 + 6
Substituting in the values of previous terms:
a4 = 3(29) − 1 + 6
a4 = 87 − 1 + 6
a4 = 92
So, the fourth term of the sequence (a4) is 92.
Finally, to find the fifth term (a5), we substitute n = 5 into the recursive formula:
a5 = 3a5 − 1 + 6
Substituting in the values of previous terms:
a5 = 3(92) − 1 + 6
a5 = 276 − 1 + 6
a5 = 281
So, the fifth term of the sequence (a5) is 281.
To summarize:
a1 = 1
a2 = 8
a3 = 29
a4 = 92
a5 = 281