Use Gauss's approach to find the following sum (do not use formulas): \( 4+9+14+19+\ldots+49 \). The sum of the sequence is \( \square \).

To find the sum \( 4 + 9 + 14 + 19 + \ldots + 49 \), we notice that the sequence increases by 5 each time. Let's rewrite the sequence in a more convenient manner: The first term is 4, and we can express the last term 49 in terms of the first term and the common difference: - The sequence can be expressed as \( a_n = 4 + (n-1) \cdot 5 \). Next, we determine how many terms there are in this sequence. We set \( a_n = 49 \) and solve for \( n \): \[ 49 = 4 + (n-1) \cdot 5 \] \[ 45 = (n-1) \cdot 5 \] \[ n-1 = 9 \implies n = 10 \] Now, we have 10 terms. To find the sum, we pair the first and last terms, moving inward. The pairs are: - \( (4 + 49), (9 + 44), (14 + 39), (19 + 34), (24 + 29) \). Each pair sums to 53, and there are 5 such pairs: \[ 5 \cdot 53 = 265 \] Thus, the sum of the sequence is \( \boxed{265} \).