Question

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. Find the number of seats in the twenty-first row.

Answer 1

Hint: Observe that the number of seats in each row is an AP with 20 being the first term 2 being the common difference. To calculate the number of seats in the twenty-first row, use the formula ${{a}_{n}}=a+\left( n-1 \right)d$, where ${{a}_{n}}$ represents the ${{n}^{th}}$ term, ‘a’ represents the first term, d represents the common difference and n represents the number of terms.

Complete step-by-step answer:
We have data regarding the number of rows and the number of seats in each row in an auditorium. We have to calculate the number of seats in the twenty-first row.
We observe that the number of seats in each row represents an AP, where the first term is the number of seats in the first row, i.e., 20 and the common difference is the difference between the number of seats in two consecutive rows.
Thus, the common difference is $=22-20=2$.
We have to calculate the number of seats in the twenty-first row.
We know that the ${{n}^{th}}$ term of an AP, whose first term is ‘a’ and the common difference is ‘d’ is given by ${{a}_{n}}=a+\left( n-1 \right)d$.
Substituting $n=21,a=20,d=2$ in the above formula, we have ${{a}_{21}}=20+\left( 21-1 \right)2$.
Simplifying the above equation, we have ${{a}_{21}}=20+20\left( 2 \right)=20+40=60$.
Hence, there are 60 seats in the twenty-first row.

Note: We observe that the number of seats increases by 2 in each subsequent row. Thus, we can also calculate the number of seats in the twenty-first row by adding 2 seats to each new row beginning from the first row. However, this is a very time-consuming method to solve the question.