The formulas shown in the image are related to arithmetic sequences. Let's break them down:

1. General Term of an Arithmetic Sequence: $u_n = u_1 + (n-1) d$

   • $u_n$ is the $n$th term of the sequence.
   • $u_1$ is the first term of the sequence.
   • $d$ is the common difference between consecutive terms.
   • $n$ is the position of the term in the sequence.

2. Sum of the First $n$ Terms of an Arithmetic Sequence (Two Forms): $S_n = \frac{n}{2} \left(2u_1 + (n-1)d\right)$

   • This is the sum of the first $n$ terms of an arithmetic sequence.
   • It uses the first term $u_1$ and the common difference $d$.

   Alternatively, $S_n = \frac{n}{2} (u_1 + u_n)$

   • This formula also gives the sum of the first $n$ terms.
   • It uses the first term $u_1$ and the $n$th term $u_n$.

How to Apply These Formulas:

Given these formulas, here’s how you might use them, depending on your problem context:

• Finding the $n$th Term ($u_n$): If you know the first term $u_1$ and the common difference $d$, you can use the first formula to find any term in the sequence.

• Calculating the Sum of the First $n$ Terms ($S_n$):

  • If you know the first term $u_1$ and the common difference $d$, use the first sum formula.
  • If you know the first and the $n$th term, use the second sum formula.

Example Application:

Suppose you have an arithmetic sequence where:

• The first term $u_1 = 5$
• The common difference $d = 3$

1. Finding the 10th term ($u_{10}$): $u_{10} = 5 + (10-1) \times 3 = 5 + 27 = 32$

2. Sum of the first 10 terms ($S_{10}$): Using the first sum formula: $S_{10} = \frac{10}{2} \left(2 \times 5 + (10-1) \times 3\right) = 5 \times (10 + 27) = 5 \times 37 = 185$

Alternatively, using the second sum formula: $S_{10} = \frac{10}{2} \times (5 + 32) = 5 \times 37 = 185$

Conclusion:

If you provide specific values (like $u_1$, $d$, $n$), I can guide you further on how to use these formulas for your problem.

Would you like to apply these formulas to a specific example or do you have any other questions?

Related Questions:

1. How do you derive the formula for the $n$th term in an arithmetic sequence?
2. Can the formulas for the sum of an arithmetic series be applied to geometric sequences?
3. What happens to the sum of an arithmetic sequence if the common difference is zero?
4. How can you determine the number of terms in an arithmetic sequence if you know the first and last terms?
5. What are some real-life applications of arithmetic sequences and series?

Tip:

When working with sequences, it's helpful to identify whether you're dealing with an arithmetic sequence (constant difference) or another type, as this will determine which formulas to use.