Hello,

Consider a geometric sequence with t3= 18 and t7= 1458. Are there one or two values for the common ratio? How does this affect the sequence

(I got one, but the way this question mentions 2 ratios, I get the feeling there are 2 values.)

My work:

1458 = 18r^4
81 = r^4
3 = r

3 answers

r = ±3

Thanks.

You are correct that there is only one value for the common ratio in this geometric sequence. Let me explain how to determine this.

To find the common ratio, we can use the formula for the nth term of a geometric sequence:

t(n) = t(1) * r^(n-1)

Here, t(n) represents the nth term, t(1) is the first term, r is the common ratio, and n is the term number.

Given that t(3) = 18, we can plug in these values into the formula:

18 = t(1) * r^(3-1)

Simplifying this equation gives us:

18 = t(1) * r^2 ---(1)

Similarly, for t(7) = 1458, we have:

1458 = t(1) * r^(7-1)

Simplifying this equation gives us:

1458 = t(1) * r^6 ---(2)

Now, we have two equations (1) and (2) with two unknowns, t(1) and r. We can solve these equations simultaneously to find the values of t(1) and r. However, we are only interested in finding the value of r, the common ratio.

Let's divide equation (2) by equation (1) to eliminate t(1):

(1458 / 18) = (t(1) * r^6) / (t(1) * r^2)

81 = r^4

Taking the fourth root of both sides, we get:

3 = r

Therefore, the common ratio for this geometric sequence is 3. This means that each term is obtained by multiplying the previous term by 3.

In conclusion, there is only one value for the common ratio, and it affects the sequence by multiplying each term by 3 to get the next term.