Let's analyze the image you provided. It asks for the following:

1. State the common difference in an arithmetic sequence.
2. Determine the missing terms in the sequence where $n = 2, 3, 4$.
3. Write the explicit formula for the sequence.

Given that:

• When $n = 1$, $f(n) = 7$.
• When $n = 5$, $f(n) = 95$.

We can calculate the common difference (d) using the formula for the arithmetic sequence: $d = \frac{f(n_5) - f(n_1)}{n_5 - n_1} = \frac{95 - 7}{5 - 1} = \frac{88}{4} = 22.$ So, the common difference $d = 22$.

Now that we know the common difference, we can fill in the missing terms:

• For $n = 2$, $f(2) = f(1) + d = 7 + 22 = 29$.
• For $n = 3$, $f(3) = f(2) + d = 29 + 22 = 51$.
• For $n = 4$, $f(4) = f(3) + d = 51 + 22 = 73$.

Finally, the explicit formula for an arithmetic sequence is given by: $f(n) = f(1) + (n - 1) \cdot d.$ Substituting the values: $f(n) = 7 + (n - 1) \cdot 22 = 7 + 22n - 22 = 22n - 15.$

Summary:

• The common difference is 22.
• The missing terms are 29, 51, and 73.
• The explicit formula is $f(n) = 22n - 15$.

Would you like more details or have any further questions?

Related Questions:

1. How is the explicit formula for an arithmetic sequence derived?
2. What is the formula for the $n$-th term in a geometric sequence?
3. How would the sequence change if the common difference was negative?
4. Can we find the sum of the first 10 terms of this sequence?
5. How can we determine if a sequence is arithmetic or geometric?

Tip:

Always check your calculated common difference by verifying multiple terms of the sequence to avoid errors.