Question

please answer this question ​

Answer 1

Given :

There is an auditorium with 35 rows of seats. There are 20 seats in the first row, 22 seats in the second row, 24 seats in the third row, and so on.

To Find :

• The number of seats in the twenty fifth row.
• Total number of seats in the auditorium.

Solution :

Analysis :

Here the concept of arithmetic progression is used. The number of seats in each subsequent rows forms an AP. The seats in twenty fifth row is the n number in AP. So by using the formula for the n number in an AP we can find the seats in twenty fifth row. And for the total number of seats is the sum of n number of AP.

Required Formula :

• aₙ = a + (n - 1)d

• $\bf S_n=\dfrac{n}{2}[2a+(n-1)d]$

Explanation :

• First term(a) = 20

• Common Difference(d) = 22 - 20 = 2

Twenty Fifth Seat :

We know that if we are given the first term and common difference of an AP and is asked to find the n number then our required formula is,

aₙ = a + (n - 1)d

where,

• aₙ = a₂₅
• a = 20
• d = 2
• n = 25

Using the required formula and substituting the required values,

⇒ aₙ = a + (n - 1)d

⇒ a₂₅ = 20 + (25 - 1)2

⇒ a₂₅ = 20 + (24)2

⇒ a₂₅ = 20 + 24 × 2

⇒ a₂₅ = 20 + 48

⇒ a₂₅ = 68

∴ a₂₅ = 68.

There are 68 seats in the twenty fifth row.

Total number of seats :

It is given that there are 35 rows in the auditorium.

So,

• n = 35

We know that if we are given the first term and common difference of an AP and is asked to find the sum n number then our required formula is,

$\bf S_n=\dfrac{n}{2}[2a+(n-1)d]$

where,

• Sₙ = S₃₅
• a = 20
• d = 2
• n = 35

Using the required formula and substituting the required values,

$\\ :\implies\sf S_n=\dfrac{n}{2}[2a+(n-1)d]$

$\\ :\implies\sf S_{35}=\dfrac{35}{2}[2(20)+(35-1)2]$

$\\ :\implies\sf S_{35}=\dfrac{35}{2}[2\times20+(34)2]$

$\\ :\implies\sf S_{35}=\dfrac{35}{2}[40+34\times2]$

$\\ :\implies\sf S_{35}=\dfrac{35}{2}[40+68]$

$\\ :\implies\sf S_{35}=\dfrac{35}{2}[108]$

$\\ :\implies\sf S_{35}=\dfrac{35}{2}\times108$

$\\ :\implies\sf S_{35}=\dfrac{35}{\not{2}}\times\cancel{108}\ \ ^{54}$

$\\ :\implies\sf S_{35}=35\times54$

$\\ :\implies\sf S_{35}=1890$

$\\ \therefore\boxed{\bf S_{35}=1,890.}$

The total number of seats in the auditorium is 1,890.