For what value of n will the nth term of the arithmetic progression 15, 12, 9, ... be equal to the nth term of the arithmetic progression -15, -13, -11, ... ?

Introduction / Context:
This question checks understanding of arithmetic progressions and the formula for the nth term. Two different arithmetic progressions are given, and we are asked to find the value of n for which their nth terms are equal. It tests algebraic manipulation and careful handling of signs.

Given Data / Assumptions:

• First arithmetic progression: 15, 12, 9, ...
• Second arithmetic progression: -15, -13, -11, ...
• The nth term of the first sequence should equal the nth term of the second sequence.
• We assume n is a positive integer.

Concept / Approach:
The nth term of an arithmetic progression with first term a and common difference d is given by a + (n - 1) * d. We find nth term expressions for both sequences and then set them equal. Solving the resulting linear equation in n gives the point where both sequences share the same term value.

Step-by-Step Solution:
Step 1: For the first progression 15, 12, 9, ... the first term a1 is 15 and the common difference d1 is 12 - 15 = -3.Step 2: The nth term of the first progression is T1(n) = 15 + (n - 1) * (-3) = 15 - 3(n - 1).Step 3: Simplify T1(n): 15 - 3(n - 1) = 15 - 3n + 3 = 18 - 3n.Step 4: For the second progression -15, -13, -11, ... the first term a2 is -15 and the common difference d2 is -13 - (-15) = 2.Step 5: The nth term of the second progression is T2(n) = -15 + (n - 1) * 2 = -15 + 2n - 2 = 2n - 17.Step 6: Set the nth terms equal: 18 - 3n = 2n - 17.Step 7: Solve for n: 18 + 17 = 2n + 3n, so 35 = 5n and n = 35 / 5 = 7.

Verification / Alternative check:
Compute the 7th term of each sequence and compare. For the first progression, T1(7) = 18 - 3 * 7 = 18 - 21 = -3. For the second progression, T2(7) = 2 * 7 - 17 = 14 - 17 = -3. Both give the same value, confirming that n = 7 is the correct solution. Checking smaller n values such as 1 or 2 quickly shows that the terms are not equal for those.

Why Other Options Are Wrong:

• n = 1: First term of first progression is 15, and of second is -15, not equal.
• n = 2: Second terms are 12 and -13, which are different.
• n = 5: Fifth terms are different and do not match -3.

Common Pitfalls:
Errors often arise from incorrect computation of the common difference or from mistakes when simplifying the nth term formula. Some students also forget to distribute the negative sign correctly in 15 - 3(n - 1). Others mistakenly equate a1 + nd instead of a1 + (n - 1)d, which shifts the sequence and leads to a wrong answer. Keeping the standard form of the nth term clear avoids these issues.

Final Answer:
The nth terms of the two arithmetic progressions are equal when n = 7.