Question 24 The 15th term of an arithmetic progression is 37, and the common difference is -2.

In an arithmetic progression, the n-th term is calculated by adding (n-1) times the common difference to the first term. By substituting the given 15th term and common difference into the formula, we can solve for the first term.

Step-by-step Derivation:
Step 1: Identify the given values from the problem.

• n = 15 (the 15th term)
• a_n = 37 (the value of the 15th term)
• d = -2 (the common difference)

Step 2: Use the general formula for the n-th term of an arithmetic progression:
a_n = a_1 + (n - 1)d

Step 3: Substitute the known values into the formula:
37 = a_1 + (15 - 1)(-2)

Step 4: Simplify the expression inside the parentheses:
37 = a_1 + (14)(-2)

Step 5: Perform the multiplication:
37 = a_1 - 28

Step 6: Solve for a_1 by adding 28 to both sides of the equation:
a_1 = 37 + 28
a_1 = 65

Final Result: The first term is 65.