S = \frac{a_1}{1 - r}

What does a negative ratio and a negative target value means in the context of this kata? Is it even specified? Does it even make any sense to expect any values?

when given a sequence and Sn (sum of the sequence) to solve for n, where n is the minimum number of terms needed to exceed (or equal) the sum of the sequence

Reference sometimes (albeit very rarely) fails at some boundary cases in random tests:

terms([2, 12, 72], 86)
3 should equal 4

Such edge cases should be added to sample tests as well.

• The a parameter is pointless if 1 is always passed in as the argument
• The output should really be a tuple with exact values
• Tests are using the old test framework
• Finding roots of a quadratic equation is a duplicate to many katas

Rounding prevents correct float comparision.

Return floats with max precision and compare with test.assert_approx_equals or math.isclose

Sample test doesn't work properly, the img in the description is dead, bad
explanation of a geometric progression from my point of view.

Returning different data types is bad practice, consider changing returning 'No Solution' for an invalid sequence to None (author seems to be inactive)

The image in the description is dead.

the given test case only test a certain index values and not random indices

getting r only works with index one and zero

Kata solution expects user to have imported math module. This should be imported by the solution, not by the user.

Please clarify that there is at least 1 root.

Duplicate.

You should add as prerequisite that all sequences tested will have more than two elements.