The number of seats in each successive row forms an arithmetic progression: 25, 29, 33,. Here, the first term is a = 25. The common difference is found by subtracting the first term from the second term: d = 29 - 25 = 4.

The number of seats in each successive row forms an arithmetic progression: 25, 29, 33,. Here, the first term is a = 25. The common difference is found by subtracting the first term from the second term: d = 29 - 25 = 4.

To find the number of seats in the 20th row, we use the formula for the n^(th) term of an arithmetic progression: a_n = a + (n - 1)d. Substituting a = 25, n = 20, and d = 4, we compute a_20 = 25 + 76 = 101.

To find the number of seats in the 20th row, we use the formula for the n^(th) term of an arithmetic progression: a_n = a + (n - 1)d. Substituting a = 25, n = 20, and d = 4, we compute a_20 = 25 + 76 = 101.

To calculate the total number of seats in 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.

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 20th row calculate the total number of seats in all 30 rows

Answer: Common difference = 4, Seats in the 20th row = 101, Total seats in 30 rows = 2490

Step-by-step solution

Step 1: Identify the arithmetic progression and common difference

The number of seats in each successive row forms an arithmetic progression: $25, 29, 33, \dots$ . Here, the first term is $a = 25$ . The common difference is found by subtracting the first term from the second term: $d = 29 - 25 = 4$ .

Step 2: Find the number of seats in the 20th row

To find the number of seats in the 20th row, we use the formula for the $n^{\text{th}}$ term of an arithmetic progression: $a_n = a + (n - 1)d$ . Substituting $a = 25$ , $n = 20$ , and $d = 4$ , we compute $a_{20} = 25 + 76 = 101$ .

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

To calculate the total number of seats in 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 $S_{30} = 15 \times (50 + 116) = 15 \times 166 = 2490$ .

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