Question

Use gauss's approach to find the following sum (do not use formulas): 3 8 13 18 ... 48

Answer 1

To find the sum using Gauss's approach, we observe the pattern and use the formula for the sum of an arithmetic sequence. The sum of the given sequence is 612.

Step-by-step explanation:

To find the sum using Gauss's approach, we need to understand the pattern in the given sequence. The sequence starts with 3, and each subsequent term is obtained by adding 5 to the previous term. So, the common difference between consecutive terms is 5. We can use the formula for the sum of an arithmetic sequence, which is Sum = (n/2)(first term + last term), where n is the number of terms.

In this case, the first term is 3 and the last term is 48. The number of terms can be found by calculating (last term - first term) / common difference + 1. Substituting the values into the formula, we get Sum = (24/2)(3 + 48) = 12(51) = 612.

Answer 2

The sum of the arithmetic sequence from 3 to 48 with a common difference of 5 is found using Gauss's approach by pairing the terms from both ends, resulting in the sum 255.

Step-by-step explanation:

Using Gauss's approach can help us find the sum of the sequence 3, 8, 13, 18, ..., 48. The sequence is arithmetic because the difference between consecutive terms is constant, which is 5 in this case. To apply Gauss's approach, we pair the numbers from either end of the sequence: (3 + 48), (8 + 43), (13 + 38), etc., until all numbers are paired. Each pair sums to 51. To find the number of terms, we look at how many 5's we add to the first term to get the last term: (48 - 3) / 5 + 1 = 10. Since we have 10 terms, that means we have 5 pairs (halve the number of terms because each pair contains 2 terms) that all sum to 51.

Now, we multiply the sum of each pair by the number of pairs: 51 × 5 = 255. Therefore, the total sum of the sequence is 255.