the 3rd term of an arithmetic sequence is 16 and the 11th term is 72 find a1, d, and the nth term

To find the first term \( a_1 \), the common difference \( d \), and the general term of the arithmetic sequence, let's start with the given information.

### Given:
- The 3rd term (\( a_3 \)) is 16.
- The 11th term (\( a_{11} \)) is 72.

### Step 1: Write the formulas for the 3rd and 11th terms.
The general formula for the \( n \)-th term of an arithmetic sequence is given by:
\[
a_n = a_1 + (n - 1)d
\]

For the 3rd term:
\[
a_3 = a_1 + 2d = 16
\]

For the 11th term:
\[
a_{11} = a_1 + 10d = 72
\]

### Step 2: Set up the system of equations.
From the above, we have two equations:
1. \( a_1 + 2d = 16 \)
2. \( a_1 + 10d = 72 \)

### Step 3: Solve the system of equations.
Subtract the first equation from the second to eliminate \( a_1 \):

\[
(a_1 + 10d) - (a_1 + 2d) = 72 - 16
\]
\[
8d = 56
\]
\[
d = \frac{56}{8} = 7
\]

Now, substitute \( d = 7 \) back into the first equation:
\[
a_1 + 2(7) = 16
\]
\[
a_1 + 14 = 16
\]
\[
a_1 = 16 - 14 = 2
\]

### Step 4: Write the nth term formula.
Now that we have \( a_1 = 2 \) and \( d = 7 \), the general formula for the \( n \)-th term of the arithmetic sequence is:
\[
a_n = a_1 + (n - 1)d
\]
Substituting the values of \( a_1 \) and \( d \):
\[
a_n = 2 + (n - 1)(7)
\]
\[
a_n = 2 + 7n - 7
\]
\[
a_n = 7n - 5So, the required values are:

- \( a_1 = 2 \) (the first term),
- \( d = 7 \) (the common difference), and
- The general term formula is \( a_n = 7n - 5 \).

If you have any questions or need further details, feel free to ask!

### Related Questions:
1. What is the 20th term of the sequence?
2. How many terms of the sequence will be needed to reach 1000?
3. What is the sum of the first 15 terms of the sequence?
4. How would the sequence change if the common difference were doubled?
5. What is the position of the term 200 in the sequence?

### Tip:
In arithmetic sequences, understanding how the common difference \( d \) affects the sequence can help quickly find terms and patterns.

Learn how to find the first term (a1), the common difference (d), and the nth term of an arithmetic sequence where the 3rd term is 16 and the 11th term is 72. This problem involves solving equations and applying the arithmetic sequence nth term formula. Suitable for Grades 9-12.

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