Math Problem Statement

How to read it all in a sentence Step 1: Identify the first term (a1a1​)

The first term of the sequence is given as 1661​.

Step 2: Determine the common difference (dd)

The common difference dd is found by subtracting the first term from the second term. For this sequence:

d=13−16=26−16=16d=31​−61​=62​−61​=61​

Verify by subtracting the second term from the third term:

12−13=36−26=1621​−31​=63​−62​=61​

So, d=16d=61​.

Step 3: Use the formula for the nth term

The formula for the nnth term of an arithmetic sequence is:

an=a1+(n−1)×dan​=a1​+(n−1)×d

Here, we want to find the 5th term (a5a5​).

Step 4: Substitute the values

Substitute a1=16a1​=61​, d=16d=61​, and n=5n=5 into the formula:

a5=16+(5−1)×16a5​=61​+(5−1)×61​a5=16+4×16a5​=61​+4×61​a5=16+46=1+46=56a5​=61​+64​=61+4​=65​

Step 5: Conclude the answer

The 5th term of the sequence is 5665​.

Solution

To summarize the process in one sentence:

To find the 5th term of the arithmetic sequence where the first term is 16\frac{1}{6} and the common difference d=16d = \frac{1}{6}, you use the formula an=a1+(n1)×da_n = a_1 + (n - 1) \times d, and after substituting n=5n = 5, a1=16a_1 = \frac{1}{6}, and d=16d = \frac{1}{6}, the 5th term is calculated as 56\frac{5}{6}.

Would you like further details or have any questions?

Here are 5 related questions to expand on this:

  1. How do you find the sum of the first nn terms of an arithmetic sequence?
  2. What happens to the common difference if the sequence is geometric instead of arithmetic?
  3. How do you determine if a sequence is arithmetic or geometric?
  4. How would you find the 100th term of this sequence?
  5. What is the formula for the nth term of a geometric sequence?

Tip: Always verify your common difference by subtracting consecutive terms; this ensures that the sequence is indeed arithmetic.

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Math Problem Analysis

Mathematical Concepts

Arithmetic Sequences

Formulas

Formula for the nth term of an arithmetic sequence: $a_n = a_1 + (n - 1) \times d$

Theorems

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Suitable Grade Level

High School