Find a_7 in the geometric series in which S_5=-61 and the common ratio is -3
In a geometric series, the sum of the first n terms can be found using the formula:
Sn = a * (1 - r^n) / (1 - r)
Where Sn is the sum of the first n terms, a is the first term, r is the common ratio, and n is the number of terms.
Given that S5 = -61 and the common ratio is -3, we can substitute these values into the formula and solve for a:
-61 = a * (1 - (-3)^5) / (1 - (-3))
Simplifying this equation:
-61 = a * (1 - 243) / 4
-61 = a * (-242) / 4
-61 = -242a / 4
-4 * -61 = -242a
244 = -242a
244 / -242 = a
-1 = a
Therefore, the first term a is -1.
To find the seventh term, we substitute the values into the formula for a geometric series:
a7 = a * r^(7 - 1)
a7 = -1 * (-3)^(6)
a7 = -1 * 729
a7 = -729
The seventh term of the geometric series is -729.