Determine the AP whose 3rd term is 5 and the 7th term is 9.
Step-by-Step Solution
Key Concept: Use the formula for the $n^{\text{th}}$ term of an arithmetic progression: $a_n = a + (n-1)d$, where $a$ is the first term and $d$ is the common difference. Set up equations for the given terms and solve the simultaneous equations to find $a$ and $d$.
1. Let $a$ be the first term and $d$ be the common difference of the AP.
2. Write the expression for the 3rd term using the AP formula:
$$a_3 = a + 2d = 5 \quad\text{(Equation 1)}$$
3. Write the expression for the 7th term:
$$a_7 = a + 6d = 9 \quad\text{(Equation 2)}$$
4. Subtract Equation 1 from Equation 2 to eliminate $a$:
$$(a + 6d) - (a + 2d) = 9 - 5$$
$$4d = 4$$
$$d = 1$$
5. Substitute $d = 1$ back into Equation 1 to find $a$:
$$a + 2(1) = 5 \Rightarrow a + 2 = 5 \Rightarrow a = 3$$
6. Hence, the required AP is:
$$\boxed{3,\;4,\;5,\;6,\;7,\;8,\;9,\;\dots}$$
(First term $a = 3$, common difference $d = 1$).
Correct Answer: First term $a = 3$, common difference $d = 1$. The AP is $3, 4, 5, 6, 7, 8, 9, \dots$.