Question

What is the first term of a geometric sequence with a common ratio (r) of 2/3 and a third term (a3) of 640/81?

A) 320/81
B) 480/81
C) 640/243
D) 960/243

Answer 1

The first term of the geometric sequence is 640/243, found by using the formula for the nth term of a geometric sequence and substituting the given third term and common ratio, then solving for the first term.

Step-by-step explanation:

To find the first term of a geometric sequence when given the third term and the common ratio, we can use the formula for the nth term of a geometric sequence, which is an = a1×r(n-1). In this case, the third term (a3) is 640/81 and the common ratio (r) is 2/3. Thus, we have:

a3 = a1×r2

640/81 = a1×(2/3)2

640/81 = a1×(4/9)

To find the first term (a1), divide both sides of the equation by 4/9:

a1 = (640/81)÷(4/9)

a1 = (640/81)×(9/4)

a1 = (640×9)/(81×4)

a1 = 5760/324

By simplifying the fraction, we find:

a1 = 640/243

Therefore, the first term of the geometric sequence is 640/243, which corresponds to option C).