The problem states:

"Determine the arithmetic progression (A.P.) whose third term is 16 and whose 7th term exceeds the 5th term by 12."

Let us solve this step-by-step:

General formula for an arithmetic progression:

The $n$-th term of an A.P. is given by: $a_n = a + (n-1)d$ where:

• $a$ is the first term,
• $d$ is the common difference,
• $n$ is the term number.

Step 1: Use the information about the 3rd term.

For the 3rd term ($a_3$) of the A.P., we know: $a_3 = a + 2d$ It is given that $a_3 = 16$. So: a + 2d = 16 \tag{1}

Step 2: Use the relationship between the 7th and 5th terms.

For the 7th term ($a_7$) and 5th term ($a_5$), we have: $a_7 = a + 6d, \quad a_5 = a + 4d$ It is given that the 7th term exceeds the 5th term by 12: $a_7 - a_5 = 12$ Substitute the formulas for $a_7$ and $a_5$: $(a + 6d) - (a + 4d) = 12$ Simplify: 2d = 12 \implies d = 6 \tag{2}

Step 3: Solve for $a$.

Substitute $d = 6$ into Equation (1): $a + 2(6) = 16$ Simplify: a + 12 = 16 \implies a = 4 \tag{3}

Step 4: Write the A.P.

The first term is $a = 4$ and the common difference is $d = 6$. The A.P. is: $4, 10, 16, 22, 28, \dots$

Final Answer:

The arithmetic progression is: $4, 10, 16, 22, 28, \dots$

Would you like a detailed explanation of any specific step? Let me know!

Related Questions:

1. What is the formula for the $n$-th term of an arithmetic progression?
2. How do you calculate the sum of the first $n$ terms of an A.P.?
3. Can you derive the common difference $d$ from two known terms of the A.P.?
4. How can you verify the consistency of the A.P. using the given conditions?
5. What is the 15th term of this A.P.?

Tip:

When solving arithmetic progression problems, always translate the conditions into equations using the general formula for the $n$-th term. This simplifies the process of finding $a$ and $d$.