If the chairs in an auditorium are arranged in an arithmetic sequence 5 th row contains 31 chair and 16 th row contains 75 chairs then find the total number of chairs if there exists 20 rows
11 differences ... 75 - 31 = 44 ... 44 / 11 = 4
difference between adjacent rows is 4
find first row and last row ... average them ... multiply by half the number of rows
Why did you post this again after you received a good answer?
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11 differences ... 75 - 31 = 44 ... 44 / 11 = 4
difference between adjacent rows is 4
find first row and last row ... add them ... multiply by half the number of rows
disregard my 1st response
Just use your formulas:
5 th row contains 31 ---> a + 4d = 31
16 th row contains 75 --> a + 15d = 75
subtract them:
11d = 44
d = 4
in a+4d = 31
a + 16 = 31
a = 15
so sum of 20 rows = (20/2)(30 + 19(4)) = ....
To find the total number of chairs in 20 rows, we need to determine the common difference in the arithmetic sequence of chairs.
Let's denote the first term of the sequence as 'a' and the common difference as 'd'.
Given that the 5th row contains 31 chairs, we can establish the following equation:
a + 4d = 31 ---(1)
Similarly, for the 16th row containing 75 chairs, we have:
a + 15d = 75 ---(2)
Now, we need to solve these two equations simultaneously to find the values of 'a' and 'd'.
Subtracting equation (1) from equation (2), we get:
11d = 44
Dividing both sides of the equation by 11, we find that d = 4.
Substituting the value of d into equation (1), we can solve for 'a':
a + 4(4) = 31
a + 16 = 31
a = 15
Now that we have the values of 'a' and 'd', we can determine the total number of chairs in 20 rows.
The formula to find the nth term of an arithmetic sequence is given by:
nth_term = a + (n - 1)d
So, to find the total number of chairs in 20 rows, we substitute n = 20:
20th_term = 15 + (20 - 1)4
20th_term = 15 + 19 * 4
20th_term = 15 + 76
20th_term = 91
Therefore, the total number of chairs in 20 rows is 91.