Finding the First and Seventh Term of an Arithmetic Progression

Arithmetic Progression (AP) is a sequence of numbers in which the difference between any two consecutive terms is constant. This constant difference is known as the common difference (d). We can represent the n-th term of an arithmetic progression using the formula:

T_n a (n-1)d

Solving for the First and Seventh Term

Let's consider the given problem where the fourth term (T4) is 39 and the sixth term (T6) is 12 more than the fourth term. We'll solve step by step to find the first and seventh terms.

Step 1: Finding the Common Difference (d)

By subtracting the first equation from the second, we get:

2d 12

Solving for d:

d 6

Step 2: Finding the First Term (a)

Substitute d 6 into the first equation:

a 3(6) 39

Simplify to find a:

a 39 - 18 21

Step 3: Finding the Seventh Term (T7)

The formula for the n-th term of an arithmetic progression is:

T_n a (n-1)d

For T7:

T7 21 6(6) 21 36 57

Alternative Methods to Verify the Solution

Using the formula directly with values:

T_7 a 6d 21 6(6) 57

Using the relationship between terms:

T_6 T_4 2d

51 39 2(6)

This verifies our solution is correct.

In summary, by solving for the common difference and the first term, we can accurately determine the seventh term of the arithmetic progression.

Note: Always ensure that the steps in solving an arithmetic progression are clear and logical. The key is finding the common difference and using the initial term to determine any other term in the sequence.