Let's start by considering the given recursive formula for the arithmetic sequence.
A(n)=A(n-1)+3; A(1)=13
To find the second, third, and fourth terms, we will substitute 2, 3, and 4 in the above formula. To do so, we will use a table.
n
A(n)=A(n-1)+3
A(n)
1
A( 1)= 13
2
A( 2)=A( 2-1)+3
A( 2)= A(1)+3 ⇕ A( 2)= 13 +3= 16
3
A( 3)=A( 3-1)+3
A( 3)= A(2)+3 ⇕ A( 3)= 16 +3= 19
4
A( 4)=A( 4-1)+3
A( 4)= A(3)+3 ⇕ A( 4)= 19 +3= 22
Therefore, the next three terms of the sequence are 16, 19, and 22. We also want to find the explicit formula of this arithmetic sequence. It combines the information provided by the two equations of the recursive form into a single equation.
Recursive:& A(n)=A(n-1)+ d;
& A(1)= A_1 [0.8em]
Explicit:& A(n)= A(1)+(n-1) d
In these formulas, d is the common difference and A(1) is the first term. Looking once again at the given recursive formula, we can identify the common difference d and the value of the first term A_1.
A(n)=A(n-1) + 3; A(1)= 13
We can see that the common difference is 3 and the first term is 13. Now we have enough information to write an explicit formula for this sequence.
