512 513 505 532 468 ?
The question asks to find the missing number in the series: 512, 513, 505, 532, 468, ?
Series Pattern Identification
Let's examine the differences between consecutive terms in the series:
- $513 - 512 = +1$
- $505 - 513 = -8$
- $532 - 505 = +27$
- $468 - 532 = -64$
The differences are $1, -8, 27, -64$. We can recognize these numbers as perfect cubes:
- $1 = 1^3$
- $-8 = (-2)^3$
- $27 = 3^3$
- $-64 = (-4)^3$
The pattern involves adding and subtracting cubes of consecutive integers starting from 1. The sequence of operations is: +$1^3$, -$2^3$, +$3^3$, -$4^3$.
Calculating the Missing Term
Following the established pattern, the next operation should be adding the cube of 5, which is $5^3$.
- The next term in the series is $468 + 5^3$.
- Calculate $5^3$: $5 \times 5 \times 5 = 125$.
- Add this to the last term: $468 + 125 = 593$.
Therefore, the number that should come in place of the question mark (?) is 593.
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