the sum of the 1st 8 terms of an AP is 80 and the sum of the next 4 terms is 88. determine the AP?
8/2 (2a+7d) = 80
12/2 (2a+11d) - 80 = 88
Now just solve for a and d.
Which formula am I going 2 use
How are am I going to do this
To determine the arithmetic progression (AP), we need to find the common difference (d) and the first term (a1).
Let's begin by using the sum formula for an arithmetic progression:
Sn = (n/2)(2a1 + (n-1)d)
where:
- Sn is the sum of the first 'n' terms of the AP,
- a1 is the first term of the AP,
- d is the common difference, and
- n is the number of terms in the AP.
Given that the sum of the first 8 terms is 80, we can substitute these values into the formula:
80 = (8/2)(2a1 + (8-1)d)
Simplifying this equation gives us:
80 = 4(2a1 + 7d)
80 = 8a1 + 28d
Next, we use the fact that the sum of the next 4 terms is 88:
88 = (4/2)(2a1 + (4-1)d)
88 = 2(2a1 + 3d)
88 = 4a1 + 6d
Now, we have a system of two equations:
80 = 8a1 + 28d (equation 1)
88 = 4a1 + 6d (equation 2)
We can solve this system of equations to find a1 and d. Subtracting equation 2 from equation 1 eliminates a1 and leaves us with:
80 - 88 = 8a1 + 28d - 4a1 - 6d
-8 = 4a1 + 22d
Simplifying further:
4a1 + 22d = -8
We can multiply this equation by 2 to eliminate the coefficients of a1:
8a1 + 44d = -16 (equation 3)
Now, we have two equations to solve:
8a1 + 28d = 80 (equation 1)
8a1 + 44d = -16 (equation 3)
By subtracting equation 1 from equation 3, we can eliminate a1:
(8a1 + 44d) - (8a1 + 28d) = -16 - 80
16d = -96
d = -6
Substituting the value of d back into equation 1:
8a1 + 28(-6) = 80
8a1 - 168 = 80
8a1 = 248
a1 = 31
Therefore, the first term (a1) of the arithmetic progression is 31, and the common difference (d) is -6.