The number of tiles in each row forms an arithmetic progression: 12, 17, 22,. The first term is a = 12. The common difference is found by subtracting the first term from the second term, giving d = 17 - 12 = 5.

The number of tiles in each row forms an arithmetic progression: 12, 17, 22,. The first term is a = 12. The common difference is found by subtracting the first term from the second term, giving d = 17 - 12 = 5.

To find the total number of tiles used in the first 18 rows, we use the formula for the sum of the first n terms: S_n = (n)/(2)[2a + (n - 1)d]. Substituting n = 18, a = 12, and d = 5, we compute S_18 = 9 × (24 + 85) = 9 × 109 = 981.

To find the total number of tiles used in the first 18 rows, we use the formula for the sum of the first n terms: S_n = (n)/(2)[2a + (n - 1)d]. Substituting n = 18, a = 12, and d = 5, we compute S_18 = 9 × (24 + 85) = 9 × 109 = 981.

School decorate a rectangular wall using tiles are used in rows the First row contains 12 tiles the second rows contains 17th tiles third row 22 tiles and the pattern continuous find the common difference how many tiles are the 18th of find the total number of tile used in the first 18 rows

Answer: Common difference = 5, Number of tiles in the 18th row = 97, Total tiles in the first 18 rows = 981

Step-by-step solution

Step 1: Identify the AP and common difference

The number of tiles in each row forms an arithmetic progression: $12, 17, 22, \dots$ . The first term is $a = 12$ . The common difference is found by subtracting the first term from the second term, giving $d = 17 - 12 = 5$ .

Step 2: Calculate the number of tiles in the 18th row

We use the formula for the $n^{\text{th}}$ term of an AP: $a_n = a + (n - 1)d$ . Substituting $a = 12$ , $n = 18$ , and $d = 5$ , we obtain $a_{18} = 12 + 17 \times 5 = 97$ . Thus, there are $97$ tiles in the $18^{\text{th}}$ row.

Step 3: Calculate the total number of tiles in the first 18 rows

To find the total number of tiles used in the first $18$ rows, we use the formula for the sum of the first $n$ terms: $S_n = \frac{n}{2}[2a + (n - 1)d]$ . Substituting $n = 18$ , $a = 12$ , and $d = 5$ , we compute $S_{18} = 9 \times (24 + 85) = 9 \times 109 = 981$ .

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