- If the 4th term of an arithmetic sequence is 24 and the 12th term is 56, whats the first term?
12
Well, let's put on our arithmetic hats and figure this out! To find the first term of an arithmetic sequence, we need to know the common difference between each term. If the 4th term is 24, we can assume the common difference is 6. How, you ask? Well, we subtract 24 and 18 from each other, and we get 6. It's like magic, but with math!
Now, armed with our common difference of 6, we can work our way backward to find the first term. If the 12th term is 56 and the common difference is 6, we can subtract 6 eleven times to land on the first term. 56 minus 66 is -10, and if that doesn't scare away the clowns, I don't know what will!
So, the first term of this arithmetic sequence is -10. Ta-da!
To find the first term of an arithmetic sequence, we can use the formula for finding the nth term of an arithmetic sequence:
Un = a + (n - 1)d
Where Un is the nth term, a is the first term, n is the position of the term in the sequence, and d is the common difference between the terms.
Given that the 4th term is 24 and the 12th term is 56, we can form the following equations:
a + (4 - 1)d = 24 -- Equation 1
a + (12 - 1)d = 56 -- Equation 2
Since we need to find the first term (a), we can solve this system of equations to find the values of a and d.
We can start by subtracting Equation 1 from Equation 2:
(a + 11d) - (a + 3d) = 56 - 24
This simplifies to:
8d = 32
Dividing both sides by 8, we get:
d = 4
Now that we know the common difference (d = 4), we can substitute this value into Equation 1 and solve for a:
a + (4 - 1) * 4 = 24
a + 3 * 4 = 24
a + 12 = 24
Subtracting 12 from both sides, we get:
a = 12
Therefore, the first term of the arithmetic sequence is 12.
a+3d = 24
a+11d = 56
subtract them:
8d = 32
d = 4
sub into one of the above equations to find a