The number of seats per row in an auditorium increases from the front to the back. the first row has 15 seats, the second row has 2 more seats than the first row, the third row has 3 more seats than the second row, the fourth row has 2 more seats than the third row, the fifth row has 3 more seats than the fourth row. this pattern continues, with successive rows alternating between 2 more seats and then 3 more seats than the previous row. how many seats are in the auditorium if there are 30 rows total?

 The required number of seats in the auditorium is 1530.

Step-by-step explanation:  Given that the number of seats per row in an auditorium increases from the front to the back.

The first row has 15 seats, the second row has 2 more seats than the first row, the third row has 3 more seats than the second row, the fourth row has 2 more seats than the third row, the fifth row has 3 more seats than the fourth row.

This pattern continues, with successive rows alternating between 2 more seats and then 3 more seats than the previous row.

We are to find the number of seats in the auditorium if there are 30 rows in total.

We know that, the sum of first n terms of an A.P. with first term a and common difference d is given by

According to the given information, we have

the number of seats in the 1st row = 15,

the number of seats in the 2nd row = 15 + 2 = 17,

the number of seats in the 3rd row = 17 + 3 = 20,

the number of seats in the 4th row = 20 + 2 = 22,

the number of seats in the 5th row = 22 + 3 =25, etc.

So, the number of sets in the odd rows is an arithmetic progression with first term 15 and common difference 5. Also, total no. of odd rows is 15.

That is, the number of seats in the odd rows is  

And, the number of sets in the even rows is an arithmetic progression with first term 17 and common difference 5. Also, total no. of even rows is 15.

That is, the number of seats in the even rows is  

Therefore, the total number of rows in the auditorium is

Thus, the required number of seats in the auditorium is 1530.

check above link.

graphically?   well, you graph them both, the solutions is where they both intersect each other, since there are two solutions, meaning they will meet twice, or namely the linear will run into the quadratic curve twice, and where they meet, those x,y coordinates are the solutions to the system.

step-by-step explanation:

this can be solved first with pythagorean theorem, which tells us to square both legs of the triangle and sum it. this will be equal to the square of the hypotenuse.

the hypotenuse is the side "opposite" the 90 degree angle, which is x.

so we can write using pythagorean theorem,

simplifying:

to further simplify this radical, we need to use the property  

thus