Find a_7 in the geometric series in which S_5=-61 and the common ratio is -3

Answer 1

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.