To find the nth term in the sequence $10, 12, 17, 19$, let's analyze the pattern.

1. Identify the differences between consecutive terms:
   • $12 - 10 = 2$
   • $17 - 12 = 5$
   • $19 - 17 = 2$

Thus, the differences alternate: $2, 5, 2$.

2. Observe the pattern: The differences alternate between $2$ and $5$.

So, the next difference after $19$ should be $5$, and then $2$ again.

Generalizing the nth term:

Let's consider the sequence in blocks of two terms, where:

• For odd-indexed terms, the difference is $+2$.
• For even-indexed terms, the difference is $+5$.

We can express the $n$-th term depending on whether $n$ is even or odd:

• For odd $n$, the term is increasing by 2 from the previous odd term.
• For even $n$, the term is increasing by 5 from the previous even term.

Formula for the nth term:

Let's divide the sequence into two cases:

1. For odd $n$ (first, third, fifth terms, etc.): $a_n = 10 + 7 \times \left(\frac{n-1}{2}\right)$

2. For even $n$ (second, fourth, sixth terms, etc.): $a_n = 12 + 7 \times \left(\frac{n-2}{2}\right)$

This captures the alternating pattern of differences.

Would you like more detailed steps on deriving this formula or have any questions?

Related questions:

1. What is the 10th term of this sequence?
2. How do we derive the general formula for arithmetic and alternating sequences?
3. What is the sum of the first 15 terms of this sequence?
4. Can we express this sequence using a recursive formula?
5. How does this sequence compare to standard arithmetic progressions?

Tip:

For sequences with alternating differences, look for patterns in how the differences change between terms to help identify the general term.