the fifth elements in an arithmetic sequence is -1 and it's twenty first element is 11. find the value of n so that a(n)= 20

To solve this problem, let's break it down step by step.

### Given Information:
- The fifth element of the arithmetic sequence is \( a_5 = -1 \).
- The twenty-first element of the arithmetic sequence is \( a_{21} = 11 \).
- We need to find \( n \) such that \( a_n = 20 \).

### Step 1: Use the General Formula for an Arithmetic Sequence
The general formula for the \( n \)-th term of an arithmetic sequence is:
\[
a_n = a_1 + (n - 1) \cdot d
\]
where:
- \( a_n \) is the \( n \)-th term,
- \( a_1 \) is the first term,
- \( d \) is the common difference.

### Step 2: Set Up Equations Using the Given Information

For the fifth element (\( a_5 \)):
\[
a_5 = a_1 + (5 - 1) \cdot d = a_1 + 4d = -1
\]

For the twenty-first element (\( a_{21} \)):
\[
a_{21} = a_1 + (21 - 1) \cdot d = a_1 + 20d = 11
\]

### Step 3: Solve the System of Equations

We have two equations:
\[
\begin{aligned}
1. \quad a_1 + 4d &= -1 \\
2. \quad a_1 + 20d &= 11
\end{aligned}
\]

Subtract equation 1 from equation 2 to eliminate \( a_1 \):
\[
(a_1 + 20d) - (a_1 + 4d) = 11 - (-1)
\]
\[
16d = 12
\]
\[
d = \frac{12}{16} = \frac{3}{4}
\]

Now substitute \( d = \frac{3}{4} \) back into the first equation:
\[
a_1 + 4\left(\frac{3}{4}\right) = -1
\]
\[
a_1 + 3 = -1
\]
\[
a_1 = -4
\]

### Step 4: Find \( n \) such that \( a_n = 20 \)

Use the general formula again:
\[
a_n = a_1 + (n - 1) \cdot d = 20
\]
Substitute \( a_1 = -4 \) and \( d = \frac{3}{4} \):
\[
-4 + (n - 1) \cdot \frac{3}{4} = 20
\]
Add 4 to both sides:
\[
(n - 1) \cdot \frac{3}{4} = 24
\]
Multiply both sides by \( \frac{4}{3} \) to solve for \( n - 1 \):
\[
n - 1 = 24 \cdot \frac{4}{3} = 32
\]
Finally, add 1 to both sides:
\[
n = 33
\]

### Final Answer:
The value of \( n \) such that \( a_n = 20 \) is \( \boxed{33} \).

Would you like more details on this solution, or do you have any questions?

### Related Questions:
1. What is the first term (\( a_1 \)) and common difference (\( d \)) in an arithmetic sequence if two non-consecutive terms are known?
2. How do you determine the sum of the first \( n \) terms of an arithmetic sequence?
3. If the 7th term of an arithmetic sequence is 2, and the 15th term is 18, what is the common difference?
4. How can you find the middle term in an arithmetic sequence with an odd number of terms?
5. If the first term and common difference of an arithmetic sequence are both positive, what can you say about the behavior of the sequence?

### Tip:
In arithmetic sequences, the difference between any two terms is constant, making it easier to find any term if you know the first term and the common difference.

Learn how to find the value of n in an arithmetic sequence where the nth term is 20. Detailed step-by-step solution using the arithmetic sequence formula. Suitable for high school students.

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