whats the first term if the third term is 21 and the eighth term is 56?
To find the first term in a arithmetic sequence, we need to determine the common difference (d) between the terms. Once we know the common difference, we can use the formula for the nth term of an arithmetic sequence to calculate the first term (a).
First, let's find the common difference:
The third term is given as 21, and the eighth term is given as 56.
Using the formula for the nth term of an arithmetic sequence:
aₙ = a + (n-1)d
Let's substitute the given values into the formula:
21 = a + (3-1)d ---(1)
56 = a + (8-1)d ---(2)
Now, we have a system of equations:
Equation (1): a + 2d = 21
Equation (2): a + 7d = 56
We can solve this system of equations to find the values of a and d:
Subtracting equation (1) from equation (2) (eliminating a):
(a + 7d) - (a + 2d) = 56 - 21
5d = 35
Dividing both sides by 5:
d = 7
Now that we have found the value of d, we can substitute it into equation (1) to find the value of a:
a + 2d = 21
a + 2(7) = 21
a + 14 = 21
Subtracting 14 from both sides:
a = 21 - 14
a = 7
Therefore, the first term of the arithmetic sequence is 7.