In an arithmetic progression, seven times the seventh term is equal to eleven times the eleventh term. What is the value of the eighteenth term of this arithmetic progression?

Introduction / Context:
This question tests your understanding of arithmetic progressions and how to use general term formulas to solve equations involving specific terms. The condition that seven times the seventh term equals eleven times the eleventh term leads to a relationship between the first term and the common difference. Once this relationship is established, you can determine the value of any term, in particular the eighteenth term.

Given Data / Assumptions:

• The sequence is an arithmetic progression, often abbreviated as AP.
• The seventh term is denoted by T7, and the eleventh term by T11.
• The condition given is 7 × T7 = 11 × T11.
• We must find the eighteenth term, T18, of the AP.
• The first term and common difference are real numbers.

Concept / Approach:
In an arithmetic progression, the nth term is given by Tn = a + (n − 1) * d, where a is the first term and d is the common difference. Using this formula, we can write expressions for T7 and T11, substitute them into the given condition, and solve for a in terms of d. Then we use the relationship between a and d to find T18 = a + 17 * d. Interestingly, in this problem the exact values of a and d are not needed, only their relationship.

Step-by-Step Solution:
Step 1: Let a be the first term and d be the common difference of the AP. Step 2: The seventh term is T7 = a + 6d. Step 3: The eleventh term is T11 = a + 10d. Step 4: The given condition is 7 * T7 = 11 * T11, so write 7 * (a + 6d) = 11 * (a + 10d). Step 5: Expand both sides: 7a + 42d = 11a + 110d. Step 6: Rearrange to group like terms. Move 7a to the right and 110d to the left: 42d − 110d = 11a − 7a. Step 7: Simplify: −68d = 4a. Step 8: Divide both sides by 4 to obtain a = −17d. Step 9: Now find the eighteenth term T18 = a + 17d. Step 10: Substitute a = −17d into T18: T18 = −17d + 17d = 0. Step 11: Therefore, the eighteenth term of the progression is 0.

Verification / Alternative check:
To verify, choose a convenient value for d and then compute a and the relevant terms. For example, let d = 1. Then a = −17. The seventh term is a + 6d = −17 + 6 = −11, and the eleventh term is a + 10d = −17 + 10 = −7. Check the condition: 7 * (−11) = −77, and 11 * (−7) = −77. The condition holds. Now compute the eighteenth term: T18 = a + 17d = −17 + 17 = 0, which matches our general result. Any other non zero value of d would scale all terms but keep T18 equal to zero because of the relationship a = −17d.

Why Other Options Are Wrong:
Values such as 1, 2, or −1 would correspond to different relationships between a and d that do not satisfy the original condition 7 * T7 = 11 * T11. When the condition is applied correctly, the only possible value of T18 is 0. The option 3 also fails for the same reason. These options may tempt students who miscompute the algebra or misapply the general term formula.

Common Pitfalls:
A common error is to miswrite the general term as a + nd instead of a + (n − 1) * d, which shifts all term indices and leads to incorrect expressions for T7 and T11. Another pitfall is mishandling the algebra when solving 7a + 42d = 11a + 110d, perhaps by incorrectly moving terms across the equality or simplifying incorrectly. Careful algebraic work and a quick numerical check with a specific choice of d help avoid these mistakes.

Final Answer:
The eighteenth term of the arithmetic progression is 0.