The sequences in Exercises 13–18 are defined using recursion formulas. Write the first four terms of each sequence. a₁=7 and a_n=a_n-1 + 5 for n≥2

Verified step by step guidance

1. Identify the initial term of the sequence, which is given as $a{1}_{}=7$.
2. Understand the recursive formula: $a{n}_{}=a{\mathrm{n-1}}_{}+5$ for $n\ge 2$. This means each term is 5 more than the previous term.
3. Calculate the second term by substituting $n=2$ into the recursive formula: $a{2}_{}=a{1}_{}+\; 5$. Use the value of $a{1}_{}$ from step 1.
4. Calculate the third term by substituting $n=3$ into the recursive formula: $a{3}_{}=a{2}_{}+\; 5$. Use the value of $a{2}_{}$ from step 3.
5. Calculate the fourth term by substituting $n=4$ into the recursive formula: $a{4}_{}=a{3}_{}+\; 5$. Use the value of $a{3}_{}$ from step 4.

Key Concepts

Recursive Sequence Definition

A recursive sequence is defined by specifying the first term(s) and a formula that relates each term to one or more previous terms. In this problem, the sequence starts with a_1 = 7, and each subsequent term is found by adding 5 to the previous term.

Sequence Terms Calculation

To find terms in a recursive sequence, start with the initial term and repeatedly apply the recursive formula. For example, given a_1 = 7 and a_n = a_(n-1) + 5, calculate a_2 by adding 5 to a_1, then use a_2 to find a_3, and so on.

Notation and Indexing in Sequences

Sequences use subscript notation (a_n) to denote the nth term. Understanding that n represents the position in the sequence and that the formula applies for n ≥ 2 is essential for correctly computing terms and interpreting the problem.