In the number series 7, 26, 63, 124, 215, 342, ( ... ), each term can be expressed as one less than a perfect cube. What is the missing next term?

Introduction / Context:
This question is about recognizing a pattern in a number series and using that pattern to determine the next term. Many competitive exams include series where each term is related to powers or cubes of integers. Here, the numbers are close to perfect cubes, and noticing this will quickly lead you to the correct answer. Being able to see such relationships is important for number series questions.

Given Data / Assumptions:

• The given series is 7, 26, 63, 124, 215, 342, ( ... ).
• We are asked to find the next term in this sequence.
• The differences between consecutive terms do not form a simple arithmetic sequence.
• We suspect a relationship to perfect cubes or some other polynomial pattern.

Concept / Approach:
Start by checking whether each term is close to a perfect cube. If we examine 2^3, 3^3, 4^3, and so on, we see that 2^3 = 8, 3^3 = 27, 4^3 = 64, 5^3 = 125, 6^3 = 216, and 7^3 = 343. Comparing these cubes to the series terms, we notice that each term is one less than the corresponding cube. Once this pattern is clear, we can predict that the next term will be one less than 8^3.

Step-by-Step Solution:
Check the first term: 7. Since 2^3 = 8, we have 8 - 1 = 7. Check the second term: 26. Since 3^3 = 27, we have 27 - 1 = 26. Check the third term: 63. Since 4^3 = 64, we have 64 - 1 = 63. Check the fourth term: 124. Since 5^3 = 125, we have 125 - 1 = 124. Check the fifth term: 215. Since 6^3 = 216, we have 216 - 1 = 215. Check the sixth term: 342. Since 7^3 = 343, we have 343 - 1 = 342. The pattern is that the nth term is n + 1 cubed minus 1, starting from n = 1. To find the next term, take 8^3 = 512. Subtract 1 to match the pattern: 512 - 1 = 511. Therefore, the missing next term in the series is 511.

Verification / Alternative Check:
Another way to verify the pattern is to subtract consecutive terms and see if those differences also follow a pattern. The differences are 26 - 7 = 19, 63 - 26 = 37, 124 - 63 = 61, 215 - 124 = 91, and 342 - 215 = 127. These difference values are 19, 37, 61, 91, 127. While these are not in a simple arithmetic progression, if we examine them closely they grow in a way consistent with the difference of cubes formula. However, the direct observation that each term is one less than a perfect cube is simpler and more reliable. Since the pattern holds for all given terms, extending it to 8^3 - 1 is well justified.

Why Other Options Are Wrong:

• Option 391: This number is not close to 8^3 and does not fit the cube minus one pattern.
• Option 421: This also does not correspond to any simple cube minus one and breaks the sequence pattern.
• Option 481: This is closer to 7^3 rather than 8^3 and does not fit the progression of increasing cubes.
• Option 573: This is far above 8^3 and completely inconsistent with the pattern of n^3 - 1.

Common Pitfalls:
Many students initially focus on the differences between successive terms and may become stuck if they do not see a simple linear pattern. Others may try to fit an arithmetic or geometric progression directly to the original terms without considering relationships to powers. It is important in number series problems to check for powers, squares, or cubes, and also to consider operations like adding or subtracting a small constant from these powers. Recognizing the 2^3, 3^3, 4^3, and so on pattern is a valuable skill.

Final Answer:
The missing next term in the series is 511.