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
rows 5 and 16 are 11 rows apart.
So, 11d = 75-31 = 44
d=4
a+4d = a+16=31, so a=15
S20 = 20/2 (2*15+19*4)
16-5=11, 75-31=44
d=4
First term = 4d-31
= 4×4-31
= -15
Number of chairs if there exists 20 rows
=n÷2(2f+(n-1)d)
=20÷2(2×(-15)+(20-1)4)
=460
To find the total number of chairs, we need to determine the common difference (d) of the arithmetic sequence.
First, we can find the 5th term of the sequence using the given information. The 5th term, a₅, is equal to 31.
Using the arithmetic sequence formula: aₙ = a₁ + (n - 1) d
where aₙ represents the nth term, a₁ is the first term, n is the number of terms, and d is the common difference.
a₅ = a₁ + (5 - 1) d
31 = a₁ + 4d
Next, we can find the 16th term of the sequence. The 16th term, a₁₆, is equal to 75.
Using the same formula as above, we have:
a₁₆ = a₁ + (16 - 1) d
75 = a₁ + 15d
We now have a system of two equations with two variables:
31 = a₁ + 4d
75 = a₁ + 15d
To solve this system, we can subtract the first equation from the second equation:
75 - 31 = (a₁ + 15d) - (a₁ + 4d)
44 = 11d
Dividing both sides by 11, we find that d = 4.
Now that we have the common difference, we can find the number of chairs in the 20th row using the formula:
a₂₀ = a₁ + (20 - 1) d
a₂₀ = a₁ + 19d
To find a₁, we can use the formula for the 5th term:
31 = a₁ + 4d
Rearranging this equation, we have:
a₁ = 31 - 4d
a₁ = 31 - 4(4)
a₁ = 31 - 16
a₁ = 15
Now, we can substitute the values into the formula for the 20th term:
a₂₀ = 15 + 19(4)
a₂₀ = 15 + 76
a₂₀ = 91
Thus, the number of chairs in the 20th row is 91.
To find the total number of chairs, we can use the formula for the sum of an arithmetic sequence:
Sₙ = (n/2)(a₁ + aₙ)
where Sₙ represents the sum, n is the number of terms, a₁ is the first term, and aₙ is the last term.
For 20 rows, the last term is a₂₀, which we found to be 91.
S₂₀ = (20/2)(15 + 91)
S₂₀ = (10)(106)
S₂₀ = 1060
Therefore, the total number of chairs in 20 rows is 1060.