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Mathematics

In a flower bed there are 43 rose plants in the first row, 41 in the second, 39 in the third and so on. There are 11 rose plants in the last row. How many rows are there and how many rose plants are there in the bed?

AP

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Answer

Given,

The number of rose plants in each row forms an Arithmetic Progression (A.P.):

43, 41, 39,……., 11.

a = 43

l = 11

d = 43 - 41 = -2

We know that,

∴ an = a + (n - 1)d

⇒ 11 = 43 + (n - 1)(-2)

⇒ 11 - 43 = (n - 1)(-2)

⇒ -32 = (n - 1)(-2)

322\dfrac{-32}{-2} = (n - 1)

⇒ n - 1 = 16

⇒ n = 16 + 1

⇒ n = 17

We know that,

Sum of n terms of an A.P. is given by,

∴ Sn = n2\dfrac{n}{2}(a + l)

Total number of rose plants:

⇒ S17 = 172\dfrac{17}{2}(43 + 11)

= 172\dfrac{17}{2}(43 + 11)

= 15 × 542\dfrac{54}{2}

= 17 × 27

= 459.

Hence, number of rows = 17, total number of rose plants in the bed is 459.

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