Let's start by considering the given recursive formula for the arithmetic sequence.
A(n)=A(n-1)- 2; A(1)=0
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)-2
A(n)
1
A( 1)= 0
2
A( 2)=A( 2-1)-2
A( 2)= A(1)-2 ⇕ A( 2)= 0 -2= - 2
3
A( 3)=A( 3-1)-2
A( 3)= A(2)-2 ⇕ A( 3)= - 2 -2= - 4
4
A( 4)=A( 4-1)-2
A( 4)= A(3)-2 ⇕ A( 4)= - 4 -2= - 6
Therefore, the next three terms of the sequence are - 2, - 4, and - 6. 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) - 2; A(1)= 0
We can see that the common difference is - 2 and the first term is 0. Now we have enough information to write an explicit formula for this sequence.
