Consider the arithmetic sequence −20, −16, −12, −8, … in which each term after the first is 4 greater than the preceding term. Which of the following numbers cannot appear as a term in this sequence?

Introduction / Context:
This is a basic question on arithmetic progressions (AP). You are given the first term and the common difference of a sequence and must decide whether a given number can be one of its terms. This is a common concept in aptitude exams and is solved using the general term formula of an AP and simple divisibility checks.

Given Data / Assumptions:

• Sequence: −20, −16, −12, −8, …
• Each term after the first is 4 greater than the preceding term.
• Therefore, the common difference d = +4.
• We must find which candidate among 220, 440, 668, 762 and 324 cannot be a term of this sequence.

Concept / Approach:
In an arithmetic progression with first term a and common difference d, the nth term T_n is given by:
T_n = a + (n − 1) * d Here, a = −20 and d = 4. We can either use this formula directly or use a modular argument: in this sequence every term differs from −20 by a multiple of 4, so every term is a multiple of 4.

Step-by-Step Solution:
Step 1: Write the general term: T_n = −20 + (n − 1) * 4. Step 2: Factor −20 as −20 = −4 * 5. We can see that every term is of the form −20 + 4k = 4(k − 5). Thus, every term in the sequence is divisible by 4. Step 3: Check each option for divisibility by 4: 220 ÷ 4 = 55 (an integer), so 220 could be a term. 440 ÷ 4 = 110 (an integer), so 440 could be a term. 668 ÷ 4 = 167 (an integer), so 668 could be a term. 324 ÷ 4 = 81 (an integer), so 324 could be a term. 762 ÷ 4 = 190 with remainder 2, so 762 is not divisible by 4. Step 4: Since every term of the sequence must be divisible by 4, any number not divisible by 4 cannot be a term. Therefore, 762 cannot be in the sequence.
Verification / Alternative check:
We can more formally check 762 with the general term formula. Suppose 762 is a term; then 762 = −20 + (n − 1)*4. Rearranging, (n − 1) * 4 = 782, so n − 1 = 782 / 4. Because 782 is not divisible by 4, n − 1 is not an integer and hence such an integer n does not exist. Therefore, 762 cannot be a term.

Why Other Options Are Wrong:
220, 440, 668, 324: These values all satisfy the divisibility requirement by 4, so each could correspond to some integer n in the formula T_n = −20 + (n − 1)*4. The question asks which number could not be a term, so these are not the correct choice.
Common Pitfalls:
A common mistake is to attempt to compute n explicitly for every option without noticing the simple pattern that all terms in the sequence are multiples of 4. Recognizing this pattern saves time and reduces calculation errors. Another pitfall is miscomputing divisibility by 4; always check the last two digits of the number to see whether they form a multiple of 4.

Final Answer:
Hence, the number that cannot be a term of the sequence is 762.