1. **Stating the problem:** We want to understand the sum of the first 8 natural numbers: $$1 + 2 + 3 + \dots + 7 + 8$$ and explore the approximations involving expressions like $$\sqrt{7(7+1)}$$ and formulas involving $$n = k$$ and $$n = k + 1$$. 2. **Formula for sum of first n natural numbers:** The sum of the first $$n$$ natural numbers is given by the formula: $$S = \frac{n(n+1)}{2}$$ This formula comes from pairing numbers from the start and end of the sequence. 3. **Calculate the sum for $$n=8$$:** $$S = \frac{8 \times (8+1)}{2} = \frac{8 \times 9}{2} = \frac{72}{2} = 36$$ So, $$1 + 2 + 3 + \dots + 8 = 36$$. 4. **Understanding the approximation $$\sqrt{7(7+1)}$$:** Calculate inside the root: $$7(7+1) = 7 \times 8 = 56$$ Then: $$\sqrt{56} \approx 7.48$$ This is not equal to the sum but might be used as an approximation or part of a different formula. 5. **Exploring the expressions involving $$n = k$$ and $$n = k + 1$$:** - When $$n = k$$, the sum is: $$S_k = \frac{k(k+1)}{2}$$ - When $$n = k + 1$$, the sum is: $$S_{k+1} = \frac{(k+1)(k+2)}{2}$$ These formulas show how the sum changes when increasing $$n$$ by 1. 6. **Approximation involving $$\sqrt{(n+1)(n+2)}$$:** For $$n = k$$, the expression: $$\sqrt{(k+1)(k+2)}$$ is close to the average of sums $$S_k$$ and $$S_{k+1}$$ but is not equal. It might be used to estimate or bound the sum. 7. **Summary:** - The exact sum of the first 8 numbers is 36. - The square root expressions like $$\sqrt{7(7+1)}$$ or $$\sqrt{(n+1)(n+2)}$$ are approximations or related expressions but do not equal the sum. - The formulas for $$n=k$$ and $$n=k+1$$ show how the sum changes with $$n$$. This explanation helps understand the sum and the related approximations shown in the image.