To find the common difference and the first term of an arithmetic sequence where the second term ([tex]\(a_2\)[/tex]) is 24 and the fifth term ([tex]\(a_5\)[/tex]) is 3, follow these detailed steps:
1. Identify the known terms:
- Second term ([tex]\(a_2\)[/tex]) = 24
- Fifth term ([tex]\(a_5\)[/tex]) = 3
2. Determine the number of terms between the given terms:
- The difference in indices between the fifth and second term is [tex]\(5 - 2 = 3\)[/tex].
3. Express the terms using the formula for the [tex]\(n\)[/tex]-th term of an arithmetic sequence:
The [tex]\(n\)[/tex]-th term is given by:
[tex]\[
a_n = a_1 + (n-1)d
\][/tex]
Where [tex]\(a_1\)[/tex] is the first term and [tex]\(d\)[/tex] is the common difference.
4. Set up equations based on the given information:
- For the second term:
[tex]\[
a_2 = a_1 + 1d \rightarrow 24 = a_1 + d
\][/tex]
- For the fifth term:
[tex]\[
a_5 = a_1 + 4d \rightarrow 3 = a_1 + 4d
\][/tex]
5. Solve the system of equations to find the common difference [tex]\(d\)[/tex]:
- Subtract the equation for [tex]\(a_2\)[/tex] from the equation for [tex]\(a_5\)[/tex]:
[tex]\[
(a_1 + 4d) - (a_1 + 1d) = 3 - 24
\][/tex]
This simplifies to:
[tex]\[
3d = -21 \rightarrow d = \frac{-21}{3} \rightarrow d = -7
\][/tex]
6. Find the first term [tex]\(a_1\)[/tex] using the value of [tex]\(d\)[/tex]:
- Substitute [tex]\(d = -7\)[/tex] back into the equation for [tex]\(a_2\)[/tex]:
[tex]\[
24 = a_1 + (-7)
\][/tex]
Solve for [tex]\(a_1\)[/tex]:
[tex]\[
a_1 = 24 + 7 \rightarrow a_1 = 31
\][/tex]
So, the common difference [tex]\(d\)[/tex] is [tex]\(-7\)[/tex], and the first term [tex]\(a_1\)[/tex] is [tex]\(31\)[/tex].
