[HELP? DUE TOMORROW] There are 10 rows in the center of the Civic Auditorium. Each row has 2 more seats than the previous row. If there are

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[HELP? DUE TOMORROW] There are 10 rows in the center of the Civic Auditorium. Each row has 2 more seats than the previous row. If there are 490 seats total in the center, how many seats are in the first row?

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    2021-09-06T01:41:38+00:00

    Answer:

    There are 40 seats in the first row

    Step-by-step explanation:

    * Lets explain how to solve the problem

    – There are 10 rows in the center of the Civic Auditorium

    – Each row has 2 more seats than the previous row

    – There are 490 seats total in the center

    * Lets solve the problem

    ∵ Each row has 2 more seats than the previous row

    ∴ There is a constant difference between each two consecutive rows

    – That means we can consider the number of seats in each row

     formed an arithmetic series

    ∵ In arithmetic series if the first term is a and the constant difference

      is d then any term in the series can get by the rule an = a + (n – 1)d ,

      where n is the position of the number in the series

    ∴ a1 = a , a2 = a + d , a3 = a + 2d , a4 = a + 3d , …….

    – The sum of n terms in the arithmetic series is Sn = n/2 [a + l] where

      l is the last term in the series

    ∵ There are 10 rows

    ∴ n = 10

    ∵ each row has 2 more seats than the previous row

    ∴ d = 2

    ∵ The first term is a

    ∴ The last term l = a + (10 – 1)d

    ∴ The last term l = a + 9d

    ∴ The last rerm l = a + 9(2) = a + 18

    ∵ The total seat in the 10s rows is 490 seats

    ∵ Sn = n/2 [a + l]

    ∴ 490 = 10/2 [a + a + 18] ⇒ simplify

    ∴ 490 = 5 [2a + 18]

    – Divide both sides by 5

    ∴ 98 = 2a + 18

    – Subtract 18 from both sides

    ∴ 80 = 2a

    – Divide both sides by 2

    ∴ a = 40

    ∵ a is the number of seats in the first row

    ∴ There are 40 seats in the first row

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