A unique kind of sequence called a geometric sequence has a constant ratio between every two succeeding terms. This ratio is regarded as one of the geometric sequence's common ratios.

In other words, each phrase in a geometric series is multiplied by the a constant to produce the following term.
Therefore, a geometric series has the formula a, ar, ar², where an is the initial term as well as r is the sequence's common ratio.
Either one positive or negative integer can be used to describe the common ratio.

Now, according to the question;

Consider right angled triangle ΔABC ; right angled at C.

The side opposite to each vertices A,B,C are a, b, c respectively.

Thus, by Pythagorean theorem,

a² + b² = c² (equation 1)

By geometric sequence;

a² = bc (say);

Also, a/c = √(b/c)

substitute in equation 1

b² + bc - c² = 0

Divide equation by c².

b²/c² + b/c - 1 = 0 (equation 2)

Consider vertex B.

The sine of angle B; sinB = Perpendicular/Hypotenuse

SinB = b/c = t (say)

Substitute b/c by t in equation 2

t² + t -1 = 0

Calculate the roots of the equation by quadratic formula;

t = (√5 - 1)/2 and (-√5 - 1)/2 (negative value is not possible for side)

Thus, t = (√5 - 1)/2

Also SinA = a/c = √(b/c)

SinA = √{(√5 - 1)/2}

Therefore, the sine of the smallest of the angles in the triangle is;

SinA = √{(√5 - 1)/2}.

To know more about geometric sequence, here

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