The sum of the first n terms is given by S_n = (n)/(2)[2a + (n - 1)d]. Substituting n = 25, a = -2, and d = 4, we compute S_25 = 25 × 46 = 1150.

the system of an ap is 18 and the 9th term is 30 find the sum of the first 25 times

Answer: $1150$

Step-by-step solution

Step 1: Set up equations for given terms

In an AP, the $n$ -th term is given by the formula $a_n = a + (n - 1)d$ , where $a$ is the first term and $d$ is the common difference. Substituting $n = 6$ gives $a + 5d = 18$ , and substituting $n = 9$ gives $a + 8d = 30$ .

Step 2: Solve for the common difference $d$

Subtracting equation (1) from equation (2) eliminates $a$ . This yields $3d = 12$ , which gives $d = 4$ .

Step 3: Solve for the first term $a$

Substitute $d = 4$ into equation (1) to solve for $a$ . This gives $a + 20 = 18$ , which simplifies to $a = -2$ .

Step 4: Find the sum of the first 25 terms

The sum of the first $n$ terms is given by $S_n = \frac{n}{2}[2a + (n - 1)d]$ . Substituting $n = 25$ , $a = -2$ , and $d = 4$ , we compute $S_{25} = 25 \times 46 = 1150$ .

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