What is the first term of a geometric sequence if its third term is −3 and          its sixth term is 81?

A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.

The formula for the nth term of a geometric sequence is:

a_n = a_1 * r^(n-1)

where:

• a_n is the nth term
• a_1 is the first term
• r is the common ratio
• n is the term number

We know that the third term (a_3) is -3 and the sixth term (a_6) is 81. We can set up the following equations:

-3 = a_1 * r^(3-1) 81 = a_1 * r^(6-1)

This simplifies to:

-3 = a_1 * r^2 81 = a_1 * r^5

We can divide the second equation by the first to get:

-27 = r^3

Taking the cube root of both sides gives:

r = -3

Substituting r = -3 into the first equation gives:

-3 = a_1 * (-3)^2 -3 = a_1 * 9 a_1 = -3/9 a_1 = -1/3

So, the first term of the geometric sequence is -1/3.