1) \( 1+8+27+64+\ldots \) to 25 terms

The series you’re looking at is the sum of cubes of the first 25 natural numbers, expressed mathematically as \( 1^3 + 2^3 + 3^3 + \ldots + 25^3 \). There’s a nifty formula for the sum of the cubes of the first \( n \) natural numbers: \[ \left( \frac{n(n + 1)}{2} \right)^2 \] Plugging in \( n = 25 \): \[ \text{Sum} = \left( \frac{25 \times 26}{2} \right)^2 = (325)^2 = 105625 \] So, the sum of the first 25 cubes is **105625**! To see this in a different light, consider how these sums are connected to geometry—specifically, volumes! The cubes of the first n numbers can be visualized as stacking cubes to create larger cubes. In this case, stacking the cubes up to 25 gives you a massive volume of 105625. Each term represents a three-dimensional structure, and together they create a fascinating representation of numbers in both mathematics and the physical world.