Sequence Term Identification

To find the 85th term of the given sequence, we first need to identify the pattern in the sequence: \(x, x^{\frac{3}{2}}, x^2, x^{\frac{5}{2}}, \dots\)

Let's look closely at the exponents of \(x\) in each term:

• For the first term, \(x\), the exponent is \(1\). We can write this as \(x^1\).
• For the second term, \(x^{\frac{3}{2}}\), the exponent is \(\frac{3}{2}\).
• For the third term, \(x^2\), the exponent is \(2\). We can write this as \(\frac{4}{2}\).
• For the fourth term, \(x^{\frac{5}{2}}\), the exponent is \(\frac{5}{2}\).

So, the sequence of exponents is \(1, \frac{3}{2}, 2, \frac{5}{2}, \dots\)

Exponents as an Arithmetic Progression

Let's check if these exponents form an Arithmetic Progression (AP). An AP is a sequence of numbers such that the difference between the consecutive terms is constant. This constant difference is called the common difference.

• Difference between the second and first term: \(\frac{3}{2} - 1 = \frac{3-2}{2} = \frac{1}{2}\)
• Difference between the third and second term: \(2 - \frac{3}{2} = \frac{4}{2} - \frac{3}{2} = \frac{1}{2}\)
• Difference between the fourth and third term: \(\frac{5}{2} - 2 = \frac{5}{2} - \frac{4}{2} = \frac{1}{2}\)

Since the difference between consecutive terms is constant (\(\frac{1}{2}\)), the exponents form an Arithmetic Progression.

For this arithmetic progression of exponents:

• The first term (\(A_1\)) is \(1\).
• The common difference (\(D\)) is \(\frac{1}{2}\).

Calculating the 85th Exponent Term

To find the 85th term of the sequence \(x, x^{\frac{3}{2}}, x^2, x^{\frac{5}{2}}, \dots\), we need to find the 85th term of the AP of its exponents.

The formula for the \(n\)-th term of an Arithmetic Progression is given by:

\(A_n = A_1 + (n-1)D\)

Here, we need to find the 85th term (\(n=85\)) of the exponent sequence:

\(A_{85} = 1 + (85-1)\left(\frac{1}{2}\right)\)

\(A_{85} = 1 + (84)\left(\frac{1}{2}\right)\)

\(A_{85} = 1 + 42\)

\(A_{85} = 43\)

So, the 85th exponent in the sequence of exponents is \(43\).

Determining the 85th Term of the Sequence

Since the original sequence is of the form \(x^{\text{exponent}}\), the 85th term of the sequence will be \(x\) raised to the power of the 85th exponent we just calculated.

Therefore, the 85th term of the sequence \(x, x^{\frac{3}{2}}, x^2, x^{\frac{5}{2}}, \dots\) is \(x^{43}\).