What is the exact value of the seventh term of the geometric progression: 686, -98, 14…?

To find the seventh term of the geometric progression, we first need to determine the common ratio. In a geometric progression, each term is obtained by multiplying the previous term by a constant factor, known as the common ratio.

From the given terms:

• The first term, a₁, is 686.

• The second term, a₂, is -98.

To find the common ratio ( r ), we divide the second term by the first term:

[

r = \frac{a₂}{a₁} = \frac{-98}{686}

]

We can simplify this fraction:

• By breaking it down, we can first find the greatest common divisor of 98 and 686. The GCD is 98, so:

[

r = \frac{-98 \div 98}{686 \div 98} = \frac{-1}{7}

]

Next, we confirm the common ratio by checking the relationship between the second and third terms:

• The third term, a₃, can be computed using the first term and the common ratio:

[

a₃ = a₂ \cdot r = -98 \cdot \left(-\frac{1}{7}\