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In the flower bed, there are 23 rose plants in the first row, twenty-one in the second row, nineteen in the third row and so on. There are five plants in the last row. How many rows are there in the flower bed?

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Hint: Find if the given series is in A.P or G.P. Find the number of terms ‘n’ in the progression using the general term formula. This will give the total number of rows in the flower bed.

The first row in the flower bed has 23 rose plants.
The second row in the flower bed has 21 rose plants.
The third row in the flower bed has 19 rose plants.
The last row in the flower bed has 5 rose plants.
So, the given sequence is 23, 21, 19,….5.
We need to find the number of rows in the flower bed which means we need to find the total number of terms in the given sequence.
To find out if the given sequence is in A.P, we need to check if the common difference between two consecutive terms in the sequence is the same for all numbers given in the sequence.
Common difference is found by taking any term and subtracting the previous term from it. We need to check the common difference for more than 1 set of consecutive numbers in the sequence. It is denoted as‘d’. If the common difference is the same throughout, then the sequence is in A.P. (Arithmetic Progression).
$
  d = 21 - 23 = - 2 \\
  d = 19 - 21 = - 2 \\
$
Since, we have the common difference equal to$ - 2$.
The first term in the A.P. is represented as ‘a’.
$a = 23,d = - 2$ …(1)
The general term of an A.P. is given by the formula $Tn = a + \left( {n - 1} \right)d$ …(2)
We need to find ‘n’ where the last term is given as 5.
So, $Tn = 5$ …(3)
Substitute (1) and (3) in (2),
$
  Tn = a + \left( {n - 1} \right)d \\
  5 = 23 + \left( {n - 1} \right)\left( { - 2} \right) \\
  n - 1\left( { - 2} \right) = - 18 \\
   - 2n + 2 = - 18 \\
   - 2n = - 20 \\
  n = 10 \\
$
So, the number of rows in the flower bed is 10.

Note: From the general term of an A.P. we can find any term’s value in the A.P. and also when the value of a term is given, we can find the term’s position in the A.P.
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