The 3rd term of an arithmetic progression is -9 and the 7th term is -29. Find the 10th term of the progression

Analysis

To solve this problem, we need to use the formula for the $n^{th}$ term of an Arithmetic Progression (AP):

Where:

• $a$ is the first term
• $d$ is the common difference
• $n$ is the position of the term

Concept

An arithmetic progression is a sequence where the difference between consecutive terms is constant. We are given the 3rd term ($T_3$) and the 7th term ($T_7$). We will use these to create a system of linear equations to find $a$ and $d$, then use those values to find the 10th term ($T_{10}$).

Step-by-Step Solution

1. Set up the equations based on the given information:

• For the 3rd term ($n=3$): $T_3 = a + (3 - 1)d = -9$ $a + 2d = -9$ --- (Equation 1)

• For the 7th term ($n=7$): $T_7 = a + (7 - 1)d = -29$ $a + 6d = -29$ --- (Equation 2)

2. Solve the system of equations: Subtract Equation 1 from Equation 2 to eliminate $a$: $(a + 6d) - (a + 2d) = -29 - (-9)$ $4d = -29 + 9$ $4d = -20$ $d = -5$

3. Find the first term ($a$): Substitute $d = -5$ into Equation 1: $a + 2(-5) = -9$ $a - 10 = -9$ $a = -9 + 10$ $a = 1$

4. Calculate the 10th term ($T_{10}$): Using $a = 1$, $d = -5$, and $n = 10$: $T_{10} = a + (10 - 1)d$ $T_{10} = 1 + 9(-5)$ $T_{10} = 1 - 45$ $T_{10} = -44$

Conclusion

Comparing our result to the options provided, the 10th term is -44, which matches option A.

Exam Tip

When dealing with AP questions, always remember that any term $T_n$ can be expressed as $T_m + (n-m)d$. For example, you could have jumped straight to the common difference by saying $T_7 = T_3 + 4d$. This saves time during the exam!