Question

In a flower garden, there are 23 plants in the first row, 21 plants in the second row, 19 plants in the third row and so on. If there are 10 rows in that flower garden, then find the total number of plants in the last row with the help of the formula, \[{{t}_{n}}=a+\left( n-1 \right)d.\]

Answer 1

Hint: In order to find the solution of this question, we have been asked to find the number of flowers in the tenth row with the formula \[{{t}_{n}}=a+\left( n-1 \right)d\] and we will see that the number of flowers in each row will form AP. So, we will try to find the value of a, n and d to get the answer.

Complete step-by-step answer:
In this question, we have been asked to find the number of flowers in the tenth row, when it is given that there are 23 flowers in the first row, 21 flowers in the second row, 10 flowers in the third row and so on.
So, to solve this question, we have been given that we have to solve the question by the formula, \[{{t}_{n}}=a+\left( n-1 \right)d.\] Now, we have been given that the first row has 23 flowers, the second row has 21 flowers, the third row has 19 flowers. So, we can see that there is a difference of 21 – 23 = 19 – 21 = – 2 flowers between any two consecutive rows and it is decreasing in each successive row. So, we can say that the number of flowers in each row is forming an AP of common difference d = – 2. And we know that ‘a’ represents the first term. So, we can say the number of flowers in the first row, we will represent the first term. Therefore, we can say a = 23.
Now, we have been asked to find the number of flowers in the tenth row by the formula, \[{{t}_{n}}=a+\left( n-1 \right)d.\] So, we can say n = 10. And, we know that a = 23 and d = – 2. Therefore, we can say,
\[{{t}_{10}}=23+\left( 10-1 \right)\left( -2 \right)\]
\[\Rightarrow {{t}_{10}}=23+9\times \left( -2 \right)\]
\[\Rightarrow {{t}_{10}}=23-18\]
\[\Rightarrow {{t}_{10}}=5\]
Here, we have obtained the value of \[{{t}_{10}}\] as 5, that means we can say there are 5 flowers in the tenth row of the flower garden.

Note: We can verify our answer by writing the first 10 terms of AP as 23, 21, 19, 17, 15, 13, 11, 9, 7, 5, ….. and then we can say that at the tenth position, we receive 5 as the answer and according to the formula also, we have obtained 5 as the answer. Therefore, our answer is correct.