The first five terms of a number sequence are shown below.10,14,18,22,26Find an expression for the (𝑛+1)th term, in terms of 𝑛.

1. Break Down the Problem

We need to identify the pattern in the given sequence and find a general expression for the $(n+1)$-th term of the sequence.

2. Relevant Concepts

The given sequence is: $10, 14, 18, 22, 26$

We observe the differences between each term:

• $14 - 10 = 4$
• $18 - 14 = 4$
• $22 - 18 = 4$
• $26 - 22 = 4$

This shows that the sequence increases by a constant value of 4, indicating it is an arithmetic sequence.

3. Analysis and Detail

The first term $a_1$ is 10, and the common difference $d$ is 4. The general expression for the $n$-th term of an arithmetic sequence is given by: $a_n = a_1 + (n-1)d$

Plugging in the values: $a_n = 10 + (n-1) \cdot 4$ Simplifying this expression: $a_n = 10 + 4n - 4$ $a_n = 4n + 6$

To find the $(n+1)$-th term, we simply plug in $n+1$: $a_{n+1} = 4(n+1) + 6$ $a_{n+1} = 4n + 4 + 6$ $a_{n+1} = 4n + 10$

4. Verify and Summarize

We confirm that this expression produces the correct terms for the sequence. For $n = 1$, $a_{2} = 4 \cdot 1 + 10 = 14$; for $n = 2$, $a_{3} = 4 \cdot 2 + 10 = 18$, and so forth.

Final Answer

The expression for the $(n+1)$-th term in terms of $n$ is: $a_{n+1} = 4n + 10$