Find a1 for the arithmetic series with Sn = 77, an=30, and n-7
Just as in your other post,
Sn = n/2 (a1 + an)
Only this time we want a1
77 = 7/2 (a1 + 30)
154 = 7a1 + 210
-56 = 7a1
a1 = -8
To find the value of a1 (the first term) in an arithmetic series, we can use the formula for the nth term of an arithmetic sequence:
an = a1 + (n - 1)d
Where:
an = nth term
a1 = first term
n = number of terms
d = common difference
In this case, we are given that Sn (the sum of the series) is 77 and we need to find a1.
The formula for the sum of an arithmetic series is:
Sn = (n/2)(a1 + an)
Given that Sn = 77, an = 30, and n = 7, we can substitute these values into the sum formula:
77 = (7/2)(a1 + 30)
To solve for a1, let's rearrange the equation:
77 = (7/2)(a1 + 30)
77 = (7/2)a1 + (7/2)(30)
77 = (7/2)a1 + 105
Now, let's isolate the variable a1:
(7/2)a1 = 77 - 105
(7/2)a1 = -28
To get rid of the fraction, we can multiply both sides of the equation by 2/7:
a1 = (-28)(2/7)
a1 = -8
Therefore, the first term (a1) in the arithmetic series is -8.