Let's start by considering the given recursive formula for the arithmetic sequence.
A(n)=A(n-1)- 4; A(1)=8
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)-4
A(n)
1
A( 1)= 8
2
A( 2)=A( 2-1)-4
A( 2)= A(1)-4 ⇕ A( 2)= 8 - 4= 4
3
A( 3)=A( 3-1)-4
A( 3)= A(2)-4 ⇕ A( 3)= 4 - 4= 0
4
A( 4)=A( 4-1)-4
A( 4)= A(3)-4 ⇕ A( 4)= 0 - 4= - 4
Therefore, the next three terms of the sequence are 4, 0, and - 4. 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) - 4; A(1)= 8
We can see that the common difference is - 4 and the first term is 8. Now we have enough information to write an explicit formula for this sequence.
