Find the sum of the first 25 terms of the sequence. \[ 5,12,19,26, \ldots \] \[ \mathrm{S}_{25}= \]

Step 1 ：We are given an arithmetic sequence, where each term is 7 more than the previous term. The sequence starts with 5 and we are asked to find the sum of the first 25 terms.

Step 2 ：The sum of the first n terms of an arithmetic sequence can be found using the formula: \(S_n = \frac{n}{2} * (a_1 + a_n)\), where n is the number of terms, \(a_1\) is the first term, and \(a_n\) is the last term. In this case, n = 25, \(a_1\) = 5, and we need to find \(a_{25}\).

Step 3 ：To find \(a_{25}\), we can use the formula for the nth term of an arithmetic sequence: \(a_n = a_1 + (n - 1) * d\), where d is the common difference. In this case, d = 7.

Step 4 ：Substituting the given values into the formula, we get \(a_{25} = 5 + (25 - 1) * 7 = 173\).

Step 5 ：Now, we substitute n = 25, \(a_1\) = 5, and \(a_{25}\) = 173 into the sum formula to get \(S_{25} = \frac{25}{2} * (5 + 173) = 2225\).

Step 6 ：Final Answer: The sum of the first 25 terms of the sequence is \(\boxed{2225}\).