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The numbers \(a_1 \), \(a_2 \), \(a_3\), ... , \(a_n\), ... is a geometric sequence. If \(a_2 \) is three more than \(a_1 \), and \(a_3\) is seven more than \(a_2 \), what is the value of \(a_4\)? Express your answer as a common fraction.

Apr 17, 2024

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We can solve this problem by using the formula for the nth term of a geometric sequence and the information given about the first few terms.

Relate Terms with Formula:

Let r be the common ratio of the geometric sequence. We know the first term (a1) but not its specific value.

We are given that the second term (a2) is three more than the first term, so a2 = a1 + 3.

Similarly, the third term (a3) is seven more than the second term, so a3 = a2 + 7.

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

an = a1 * r^(n-1)

Find the Common Ratio (r):

We can use the information about the second and first terms to find the common ratio. We know that a2 = a1 * r:

a2 = a1 + 3 (given) a1 * r = a1 + 3

Since a1 is not zero, we can divide both sides by a1 to isolate r:

r = 1 + 3/a1

Find a3 using the formula and r:

We can now use the formula for a3 and the value we found for r:

a3 = a1 * r^2 = a1 * (1 + 3/a1)^2

We are also given that a3 = a2 + 7:

a1 * (1 + 3/a1)^2 = a1 + 3 + 7

Simplify and Solve for a1:

Expanding the square in the first term:

a1 + 6/a1 + 9/a1^2 = a1 + 10

Combining like terms:

6/a1 + 9/a1^2 = 10

Taking a common denominator of a1^2:

6a1 + 9 = 10a1^2

Rearranging the equation:

10a1^2 - 6a1 - 9 = 0

Factoring the expression:

(2a1 + 3)(5a1 - 3) = 0

Therefore, a1 = -3/2 or a1 = 3/5.

Find a4 based on a1 and r:

Since a negative value for a1 wouldn't make sense in a geometric sequence with positive terms, we know a1 = 3/5. Now we can find the common ratio (r):

r = 1 + 3/a1 = 1 + 3 / (3/5) = 8/3

Finally, let's find the value of a4 using the formula and the values of a1 and r:

a4 = a1 * r^3 = (3/5) * (8/3)^3 = (3/5) * 512/27

Therefore, the value of a4 is 1024/135, expressed as a common fraction.

Apr 17, 2024