To find the number of seats in the 23rd row, we use the n^(th) term formula a_n = a + (n - 1)d. Substituting n = 23, a = 25, and d = 4, we obtain a_23 = 25 + 88 = 113.

To find the total seats across all 30 rows, we use the sum formula S_n = (n)/(2)[2a + (n - 1)d]. Substituting n = 30, a = 25, and d = 4, we get S_30 = 15(50 + 116) = 15 × 166 = 2490.

a theatre has 25 seeds in the First row 29 in the second row 33 in the third row and so on there are 30 rows find the common difference find the number of seats in the 23 calculate the total number of seats in all 30 rows

Answer: Common difference = 4; Seats in the 23rd row = 113; Total seats in 30 rows = 2490

Step-by-step solution

Step 1: Identify the AP and common difference

The number of seats in the successive rows are $25, 29, 33, \dots$ . Since the difference between consecutive terms $29 - 25 = 33 - 29 = 4$ is constant, this list forms an Arithmetic Progression with first term $a = 25$ and common difference $d = 4$ .

Step 2: Find the number of seats in the 23rd row

To find the number of seats in the 23rd row, we use the $n^{\text{th}}$ term formula $a_n = a + (n - 1)d$ . Substituting $n = 23$ , $a = 25$ , and $d = 4$ , we obtain $a_{23} = 25 + 88 = 113$ .

Step 3: Calculate the total number of seats in all 30 rows

To find the total seats across all 30 rows, we use the sum formula $S_n = \frac{n}{2}[2a + (n - 1)d]$ . Substituting $n = 30$ , $a = 25$ , and $d = 4$ , we get $S_{30} = 15(50 + 116) = 15 \times 166 = 2490$ .

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