In an arithmetic progression, the third term is -8 and the ninth term is 10. What is the value of the sixteenth term of this progression?

Introduction / Context:
This problem is a classic arithmetic progression question where two nonconsecutive terms are given and you are asked to find another term. Such questions test your understanding of the formula for the nth term of an arithmetic progression and your ability to form and solve simple linear equations using the given information.

Given Data / Assumptions:

• Let the first term of the arithmetic progression be a.
• Let the common difference be d.
• The third term a3 is -8.
• The ninth term a9 is 10.
• We are required to find the sixteenth term a16.

Concept / Approach:
For an arithmetic progression, the nth term is given by a_n = a + (n - 1) * d. Using this formula, we can express the third term and the ninth term in terms of a and d. Since both of these terms are provided, we obtain two linear equations in a and d. Solving these equations gives us the values of a and d. Once we know the first term and the common difference, we can substitute into the term formula again to find the sixteenth term.

Step-by-Step Solution:
Write the third term using the formula: a3 = a + 2d = -8. Write the ninth term using the formula: a9 = a + 8d = 10. Subtract the first equation from the second to eliminate a: (a + 8d) - (a + 2d) = 10 - (-8). This simplifies to 6d = 18, so d = 18 / 6 = 3. Substitute d = 3 back into a + 2d = -8 to find a. Then a + 2 * 3 = -8, so a + 6 = -8. Therefore a = -8 - 6 = -14. Now find the sixteenth term using a16 = a + 15d. So a16 = -14 + 15 * 3 = -14 + 45 = 31. Hence, the sixteenth term of the arithmetic progression is 31.

Verification / Alternative Check:
We can verify our result by generating a few terms. With a = -14 and d = 3, the sequence starts as -14, -11, -8, -5, -2, 1, 4, 7, 10, ... We see that the third term is indeed -8 and the ninth term is 10, matching the data. Continuing the pattern, the tenth term is 13, the eleventh term is 16, the twelfth term is 19, the thirteenth term is 22, the fourteenth term is 25, the fifteenth term is 28, and the sixteenth term is 31. This confirms that our computed sixteenth term is correct.

Why Other Options Are Wrong:

• Option 34: This would require either a different common difference or a different first term and does not fit with the given third and ninth terms.
• Option 28: This value actually appears as the fifteenth term in the correct progression, not the sixteenth term.
• Option 25: This value matches the fourteenth term, not the sixteenth, so it is incorrect for this question.
• Option 37: This would correspond to an even larger term and is inconsistent with the established pattern of d = 3.

Common Pitfalls:
A common mistake is to misapply the nth term formula and write a_n = a + nd instead of a + (n - 1) * d. Another frequent error is to subtract the equations in the wrong order or handle negative signs carelessly, leading to an incorrect value for the common difference. Some students also try to guess the sequence without forming equations, which can lead to confusion, especially if negative terms are involved. Working systematically with equations avoids these issues.

Final Answer:
The sixteenth term of the arithmetic progression is 31.