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Step-by-step explanation:
To find the first five terms of an arithmetic sequence, we need to determine the common difference \( d \) and the first term \( a_1 \). We know the 4th term (\( a_4 \)) is -1 and the 6th term (\( a_6 \)) is 9.
In an arithmetic sequence, the \( n \)-th term is given by:
\[ a_n = a_1 + (n - 1)d \]
Using the information provided:
1. For the 4th term: \( a_4 = a_1 + 3d = -1 \)
2. For the 6th term: \( a_6 = a_1 + 5d = 9 \)
Now, we can solve these two equations to find \( a_1 \) and \( d \):
**Step 1:** Subtract the first equation from the second to find \( d \):
\[ (a_1 + 5d) - (a_1 + 3d) = 9 - (-1) \]
\[ 2d = 10 \]
\[ d = 5 \]
**Step 2:** Substitute \( d \) back into the first equation to find \( a_1 \):
\[ a_1 + 3(5) = -1 \]
\[ a_1 + 15 = -1 \]
\[ a_1 = -1 - 15 \]
\[ a_1 = -16 \]
Now, we have the first term \( a_1 = -16 \) and the common difference \( d = 5 \). Let's find the first five terms:
1. **First term (\( a_1 \))**: \(-16\)
2. **Second term (\( a_2 \))**: \( a_1 + d = -16 + 5 = -11 \)
3. **Third term (\( a_3 \))**: \( a_1 + 2d = -16 + 10 = -6 \)
4. **Fourth term (\( a_4 \))**: \( a_1 + 3d = -16 + 15 = -1 \)
5. **Fifth term (\( a_5 \))**: \( a_1 + 4d = -16 + 20 = 4 \)
So, the first five terms of the sequence are: \(-16, -11, -6, -1, 4\).