The fifth term of an arithmetic sequence is 9 and the 32nd term is -84. What is the 23rd term?
To find the 23rd term of an arithmetic sequence, we need to determine the common difference (d) first.
Given that the fifth term (a₅) is 9 and the 32nd term (a₃₂) is -84, we can use the formula:
aₙ = a₁ + (n - 1) * d
For the fifth term (n = 5):
9 = a₁ + (5 - 1) * d
9 = a₁ + 4d
For the 32nd term (n = 32):
-84 = a₁ + (32 - 1) * d
-84 = a₁ + 31d
We now have a system of two equations:
9 = a₁ + 4d
-84 = a₁ + 31d
Let's solve the system of equations:
Multiply the first equation by 31:
279 = 31a₁ + 124d
Multiply the second equation by 4:
-336 = 4a₁ + 124d
Now subtract the second equation from the first:
279 - (-336) = 31a₁ - 4a₁
615 = 27a₁
Divide both sides by 27:
a₁ = 615 / 27
a₁ = 23
Substitute the value of a₁ into the first equation to find d:
9 = 23 + 4d
4d = 9 - 23
4d = -14
d = -14 / 4
d = -3.5
Now that we have the common difference (d), we can find the 23rd term (a₂₃):
a₂₃ = a₁ + (n - 1) * d
a₂₃ = 23 + (23 - 1) * (-3.5)
a₂₃ = 23 + 22 * (-3.5)
a₂₃ = 23 - 77
a₂₃ = -54
Therefore, the 23rd term of the arithmetic sequence is -54.
To find the 23rd term of an arithmetic sequence, we need to use the formula for the nth term of an arithmetic sequence.
The formula for the nth term of an arithmetic sequence is:
nth term = a + (n - 1) * d
where "a" is the first term of the sequence, "n" is the position of the term we want to find, and "d" is the common difference between the terms.
The information given in the problem is that the fifth term is 9 and the 32nd term is -84. We can use this information to find the common difference.
Using the formula, we have:
9 = a + (5 - 1) * d => 9 = a + 4d ---(1)
-84 = a + (32 - 1) * d => -84 = a + 31d ---(2)
We now have a system of two equations with two variables (a and d). We can solve this system to find the values of a and d.
To solve the system, we can subtract equation (1) from equation (2):
(-84) - 9 = (a + 31d) - (a + 4d)
-93 = 27d
Dividing both sides of the equation by 27, we have:
-93/27 = d
-3.44... = d
Now that we know the value of the common difference (d), we can substitute it back into equation (1) or equation (2) to find the value of the first term (a).
Using equation (1):
9 = a + 4 * (-3.44...)
Simplifying, we have:
9 = a - 13.76...
Adding 13.76... to both sides:
9 + 13.76... = a
22.76... = a
Now we have found the values of both a and d.
To find the 23rd term, we can substitute these values into the formula for the nth term:
23rd term = 22.76... + (23 - 1) * (-3.44...)
Simplifying, we have:
23rd term = 22.76... + 22 * (-3.44...)
23rd term = 22.76... - 75.68...
Adding the two numbers, we have:
23rd term = -52.92...
Therefore, the 23rd term of the arithmetic sequence is approximately -52.92.